The Phenomenon of Bistable Phase Difference Intervals in the Times-Frequency Vibration Synchronization System Driven by Two Homodromy Exciters

A vibration system with two homodromy exciters operated in diﬀerent rotational speed is established to investigate whether the phenomenon of bistable phase diﬀerence intervals exists in the times-frequency vibration synchronization system. Some constructive conclusions are proposed. (1) By introducing an average angular velocity perturbation parameter ε 0 and two sets of phase diﬀerence perturbation parameters and ε 2 , the frequency capture criterion and the necessary criteria for realizing the times-frequency vibration synchronization are derived. The corresponding stability analysis is carried out. (2) By the theoretical analysis and experiments, it is veriﬁed that the times-frequency vibration synchronization system exists the phenomena of bistable phase diﬀerence interval. That is, the phase diﬀerences between the two homodromy exciters are stable around 180 degrees when they are located at a short distance; the antiphase synchronization phenomenon appears. On the contrary, they are stable around 0 degrees at the in-phase synchronization state. (3) Because of the two homodromy exciters operating in the diﬀerent rotational speed, the vibration system obtains relatively complex compound motion trajectories; the corresponding application is investigated by adding a feeding material chamber. The times-frequency vibration synchronization system can be used to design the vibration mill for reducing its low-energy zone and developing chaotic mixing equipment for obtaining a better mixing eﬀect.


Introduction
Synchronization can be explained as the adjustment of oscillating objects due to their weak interaction. As early as in the 17th century, because Huygens studied the coupling phenomenon of pendulum clocks, the phenomenon of inphase synchronization and antiphase synchronization was observed [1]. In the last century, Blekhman repeated this experiment, and the phenomenon was further observed and explained in theory. Later, Kozlov et al. found two van der Pol-Duffing oscillators with nonlinear coupling which can also realize the bistable phase synchronization [2]. e phenomena of bistable phase synchronization in nature have attracted more and more researchers to study deeply, such as active Nambu mechanics system [3], inhibitory coupled bursting neurons system [4], and coupled systems of piecewise constant oscillators [5].
Blekhman and Lurie have used the Poincare-Lyapunov small parameter method to study the vibration system composed of double eccentric rotors in 1953, pointing out that the system could realize self-synchronization and proving the system's motion stability [6,7]. Since then, the synchronization theory in the mechanical system has been studied. Recent studies have shown that the vibration system also has double phase synchronization phenomena, which mainly exist in four forms: (1) the bistable phase synchronization phenomenon is natural to appear in some vibration systems with double vibration mass or multimass [8,9], including some rotor-pendulum systems [10,11]; (2) some vibration systems have the in-phase and antiphase synchronization phenomena when they operate in a subresonance or a superresonance state, respectively [12,13]; (3) the event of bistable phase difference intervals also exists in some near-resonance nonlinear vibration synchronous systems [8,10]; (4) it is more popularly known in the vibration synchronization system with two homodromy exciters located at a single vibration mass body [14][15][16], including rotor-pendulum systems [10] and two eccentric rotors with a common rotational axis system [17,18]. e inphase synchronization phenomenon between the two homodromy exciters appears when they are located at a relatively far distance, i.e., the stable phase difference interval is (− 90°, 90°). On the contrary, the antiphase phenomenon appears and the steady interval is (90°, 270°). In this condition, the system operates in the swing state, it has little engineering application in a general way. erefore, the inphase synchronization phenomenon gains more traction compared with antiphase synchronization. Meanwhile, two or more exciters operating in the in-phase synchronization state can make the exciting force superimposition, which is possible to make the system gain a larger vibration amplitude with linear or near-elliptic motion trajectories.
In actuality, the structure of the system with two or more than two vibration masses is complex. Because of the limiting minimum natural frequency of the supporting springs, the rotational speed of the exciters is impossible to run at a relatively high speed, which makes the subresonance system not obtain a more significant exciting force even its structure is more complicated.
us, the in-phase vibration synchronization system is usually designed as a single vibration mass with two homodromy exciters located at a far distance and operates in a superresonance state. But, the more realistic question is, just as mentioned above, obtaining a stable synchronization state for a system relies on the adjustment of oscillating objects due to their weak interaction. e longer distance between the two exciters means the more weak interaction; it is unfavorable for a mechanical system. At the same time, the impulse loads from mineral and some other external factors can easily disturb the weak interaction, which makes the system change violently to deviate from the normal in-phase synchronization state.
us, for obtaining a stable synchronization state, it is necessary to design the vibration system with a relatively shorter distance between the two exciters for achieving a robustness state. Now that the vibration system powered by the same frequency supply cannot obtain the desired motion trajectories quickly, what can we do to realize the engineering requirements and protect the system from this kind of vulnerability? Maybe a times-frequency synchronization system is the right choice. Some vibration machines rely on gear unit or other speed change mechanism unit to achieve the synchronization state, such as double-frequency vibration compacting machine, vibrating feeder, multiaxis inertia table concentrator, and screening machine [19,20]. e forced synchronization by rigid coupling causes even more serious problems especially in cracks of the mechanical structure, makes the system more complex, and decreases system reliability and stability. erefore, the timesfrequency vibration synchronization theory may take on even more importance.
By the symmetric layout of four exciters in a vibration system of plane motion, the phenomenon of tripling frequency vibration synchronization is observed and investigated by Inoue and Araki in the 1970s [21]. It is a pity that the study did not continue to carry out related research and experiments on this basis. Subsequently, in the 1980s, Wen had pointed out that some nonlinear systems could achieve the times-frequency synchronization state. e system designed by Wen has two vibration masses; each one has two degrees of freedom, supported by piecewise nonlinear springs [19]. On the one hand, this system has a relatively complex mechanical structure; on the other hand, it rests in the conceptual phases of product design and the reliability cannot be proved by experiments. e difficulty in the design of the times-frequency vibration synchronization system is just as the one pointed out by Wen; it is more challenging to implement the high-order harmonics and the subharmonic times-frequency capture than the fundamental frequency capture because of the smaller frequency capture interval [19]. Recently, a timesfrequency vibration system has been investigated by Jia, who gets a stable state by way of controlled synchronization [22]. If we can control the system based on the optimal structure parameters of the mechanical model, it will raise the efficiency, enhance the stability and the robustness, and decrease the energy loss by forcing control of the system. us, it is necessary and urgent to investigate the times-frequency vibration system to develop the theory of times-frequency synchronization and give out the critical structural parameters.
In our previous research, a vibration system with two exciters rotating in opposite directions is investigated. e condition of times-frequency vibration synchronization and its stability criterion are obtained, which is also proved by experimental studies. It shows that the system only exhibits in-phase vibration synchronization phenomenon [23]. In this study, we mainly investigate the vibration system with two homodromy exciters operating at different rotational speeds and find out whether the system exhibits bistable phase difference intervals and gives out the critical structural parameters and corresponding criteria. Furthermore, the engineering application of times-frequency vibration synchronization is also explored by adding a feeding material chamber to observe the motion trajectories of particle materials.

Dynamics Equation and Steady-State
Response of the System

Dynamics Equation of the
System. e typical model of the vibration synchronization system exciting by two homodromy exciters is shown in Figure 1, where oxy is the fixed coordinate system, o ′ x ′ y ′ is the moving coordinate system, m is the mass of the rigid frame, o is the mass center of the whole vibration system, k x , k y and c x , c y are the stiffness and damping of the system in x, y direction, respectively. m i and r i are the mass and eccentric radius of the eccentric block of the exciter i(i � 1, 2), respectively. _ φ i and φ i are the instantaneous angular velocity and the angle of rotation relative to the starting point of the exciter i(i � 1, 2), respectively. l i and β i are the distance from the axis of the exciter i(i � 1, 2) to the mass center o of the system and the angle between its connection and the horizontal direction, respectively. l si and θ i (i � 1, 2) are the distance from the connecting point between the spring i(i � 1, 2) and frame to point o and the angle between its connection and the horizontal direction. l 0 is the distance from the composite mass center of the two exciters to the mass center of the system, β 0 is the angle between l 0 and the horizontal direction, and l is the distance from the axis of exciter 1 to the axis of exciter 2. By the way, all the nomenclature of the physical parameters and the symbols of intermediate parameters are normalized and listed at the end of the article and Table 1, respectively.
Considering that the angular displacement ψ of the system is too small comparing with β i and θ i (i � 1, 2), o can be left out o when calculating the kinetic energy of the system and the potential of the springs. Based on Figure 1, the equation of kinetic energy, potential energy, and energy dissipate function of the system is given as follows: where J p is the moment of inertia of the vibration system to its mass center o, J i is the moment of inertia of the eccentric mass of the exciter i(i � 1, 2) to its mass center, and c i is the damping coefficient of the axes of the exciter i(i � 1, 2). Taking q � [x, y, ψ, φ 1 , φ 2 ] T as the generalized coordinate, substitute the kinetic energy T, potential energy V, and energy dissipate function D 0 of the system into the Lagrange equation and consider the following relations: (2) en, the differential equation of motion of the system is obtained: where J i is the moment of inertia of the exciter i(i � 1, 2) and L ei is the output torque of the motor which is driving the exciter i(i � 1, 2).

Steady-State
Response of the System. For a timesfrequency vibration system, the angular velocities of two exciters rotating in the same direction have definite multiple relations. Assume n is the multiple relationships of rotational speed between the two exciters and T 0 is the single cycle of the high-speed exciter. Considering the multiple relationships in a common period nT 0 , the angle of rotation should be 2nπ and 2π, respectively. Suppose the average phase of two exciters in steady-state is φ and the average phase difference between the two exciters is α. e phase relations in a common period between the two exciters can be represented by Figure 2 and the following equation: where the steady-state phase difference of two exciters in its every single cycle is α 1 and α 2 , respectively. From Figure 2 and equation (4), it can be concluded that the average steady-state phase difference of two exciters in a single common period is as follows: In this case, when the system is in a steady-state, the angular velocity _ φ i of the eccentric block of exciter i(i � 1, 2) should be constant; the relation between steady angular velocity and angular acceleration of the two exciters is as follows: Usually, in the same type of vibration system where the majority of models of the exciters are interchangeable, we only need to adjust the participating mass of the eccentric block to obtain the required exciting forces, that is, we can assume r 2 � r 1 � r, m 2 � ηm r , and m 1 � m r . Considering the group arrangement of the same springs and the symmetrical arrangement of the exciters, we can get the following relations: k x1 � k x2 � k x /2, k y1 � k y2 � k y /2, c x1 � c x2 � c x /2, c y1 � c y2 � c y /2, l s1 � l s2 � l s , β 1 � π − β, β 2 � β, θ 1 ≈ π, θ 2 ≈ 2π, and l 1 � l 2 � l/2 cos β. In this case, the first three equations in equation (3) can be simplified to where M is the total mass of the vibration system, M � m + 2 i�1 m i , and J is the moment of inertia of the whole vibration system to the mass center of the system, J � J p + ml 2 0 + 2 i�1 m i l 2 i . For a vibration system, the supporting positions of the spring can profoundly influence the dynamic characteristics of the whole system. As shown in equation (7), the location of the spring and the related parameters has a noticeable influence on the stiffness and damping of the system in the ψ direction when it swings around the mass center of the body. Transforming equation (7) to the dimensionless form, we can conclude that where e damping effect on the amplitude of the vibration system can be neglected because the damping ratio of the system is too small. Furthermore, the steady-state response of the system can be obtained by using the superposition principle as follows: where As mentioned in related references, it is noteworthy that the operating frequency of the superresonance vibration system is more than three times that of its natural frequency. For the multifrequency vibration machine, because of the limitation of the maximum speed of the exciting motor and the natural frequency of the supporting springs, the rotation speed of the low-speed exciter of the multifrequency vibration system can quickly fall near the natural frequency of the system. At this time, according to the phase-frequency characteristics of the vibration system, we can get that c i2 ≈ π/2, (i � x, y, ψ). e trajectories of this kind of nearresonant machine are more complex and should be avoided as much as possible.

Frequency Capture Equation of the Times-Frequency Vibration
System. e small perturbation parameter ε 0 (t) of average speed _ φ and the small phase perturbation parameters ε 1 (t) and ε 2 (t) of the two exciters are introduced when the system operates in a steady-state. In this case, the speed and acceleration of the two exciters in equation (6) can be expressed more accurately as follows:

Mathematical Problems in Engineering
According to the requirements of the last two equations in equation (3), the required derivatives in x, y, and ψ directions are easily worked out from the time-domain response of equation (10). en, the relations in equation (12) should be considered. After substituting them into equation (10), the average values are obtained by integrating them in a common period. As known from equation (3), the average angular velocity should be constant when the system operates in a steady state, that is, when calculating the average values by integrals, there must be en, the following equations can be obtained: where Because the rotational inertia of the axis of the exciter is much smaller than that of the eccentric mass on the axis, it can be neglected. us, the moment of inertia of the exciter is J i ≈ m i r 2 i (i � 1, 2). Considering the relation between the motor output torque L e0i of the exciter i(i � 1, 2) and its stiffness coefficients k e0i of the angular velocity at a certain speed ω m , e frequency capture equation of the times-frequency vibration synchronization system can be obtained by substituting them into equation (14) and sorting it out as a matrix form. We have where

Necessary Conditions for Realizing Times-Frequency
Vibration Synchronization. For the dynamic system shown in equation (3), to make the system running stability, as mentioned before, the first three equations of equation (3) must have a steady solution. Furthermore, the last two equations of equation (3) also must be stable under the perturbations, that is, for the frequency capture equation of the times-frequency vibration synchronization system shown in equation (17), there must be _ ε i � 0 and ε i � 0 (i � 1, 2). It means u i � 0 (i � 1, 2). en, subtracting the two equations, the moment balance equations for the frequency capture of two exciters can be obtained: where It is noteworthy that the system damping of the vibration synchronization equipment is small. For the vibration system which does not work in the subresonance or superresonance state, the value of phase difference angle of the time-domain displacement responses lagging the exciting loads approximates to zero in the low-frequency working condition, which is approximate to π in the high-frequency working condition. erefore, it means that W sis , W si0 , and W cis are much less than W sic , W ci0 , and W cic (i � 1, 2) in the frequency capture equation.
en, equation (19) can be simplified to Mathematical Problems in Engineering Furthermore, we have where δ � arctan and D is a synchronization performance index with a value of If |D| < 1, there are no solutions in equation (22), the stationary equation cannot be satisfied, the eccentric block of the two exciters does not have a stable phase difference, and the system cannot achieve a times-frequency synchronization vibration state. erefore, the necessary condition for realizing times-frequency vibration synchronization of the vibration system is the synchronization performance index:

Stability Criterion of the Vibration Synchronization State.
To analyze whether the system can run stably near the equilibrium point, the critical point is to check the global stability of the system at the equilibrium point. Moreover, we need to discuss whether the system is uniformly asymptotically stable and exponentially stable at the equilibrium point.

Equivalent Perturbed System.
e Taylor expansion is used to linearize equation (17) at α � α 0 and ω m � ω m0 . en, the equivalent linearized formula is obtained by taking into account that W sis , W si0 , are W cis are far less than W sic ,W ci0 , and W cic (i � 1, 2): where When E 0 is invertible, the following equation should be satisfied: For the vibration system, the mass m of the eccentric block of the exciter is far less than the mass M of the whole vibration system, so r m ≪ 1. Besides, the vibration synchronization system working in the superresonance state is a typical low damped vibration system and the phase of the time-domain response solution is almost antiphase, different from that of the exciting load. Considering r l < 7 as discussed in [24,25], we have ρ 1 ≪ 1 and ρ 2 ≪ 1 in equation (26). Furthermore, it means that Δ in equation (28) must be greater than 0.
en, a uniform model of the perturbed system of equation (26) can be given as where 8 Mathematical Problems in Engineering It is noteworthy that the remainder term g(t, x) contains α, which is dependent on the time-domain variable t, usually named as a perturbation factor of the phase difference. Although the changes of small parameters ε 1 and ε 2 will cause the change of α, the equation of the remainder term has a definite upper limiting value and lower limiting value, that is, for all t ≥ 0, there is a positive value c can make ‖g(t, x)‖ 2 < c‖x‖ 2 . In this case, according to [26], the term g(t, x) can be treated as an additive term of the system _ x � f(t, x), which does not change the order and the uncertainty of the system.

Stability of the Uniform Perturbed System under Angular Velocity
Perturbation. According to equation (29), when x � 0, g(t, 0) � 0, which means that the origin point of the perturbed system is an equilibrium point. In this case, the stability of the origin point can be analyzed as the stability of the perturbed system. at is to say, the problem of whether the times-frequency vibration system can achieve the vibration synchronization state can be directly treated as solving the stability of the system at the origin point under the condition of the angular velocity perturbation and the phase difference perturbation.
Firstly, considering the stability of the system under angular velocity perturbation parameters, i.e., let phase difference perturbation term g(t, x) � 0, the system in equation (29) becomes a two-dimensional ordinary differential system. When the matrix A satisfies the criteria required by the Hurwitz matrix, i.e., all the eigenvalues of A having Reλ(A) < 0, the system will be asymptotically stable at the origin point and the zero solutions are asymptotically stable. In this case, the first-order subdeterminants matrix of the coefficient matrix A is less than 0 and its second-order subdeterminants matrix must be greater than 0, that is, As shown in equation (17), the term mω 2 m r 2 in the denominator is the exciting force multiplied by the eccentric radius of the exciter. It is far less than the stiffness coefficients k e0i (i � 1, 2) of the angular velocity of the motor of the exciter [27]. At the same time, we note that ρ 2 ≪ 1. erefore, the criteria in equation (31) can be satisfied. Furthermore, it means that the matrix A is a Hurwitz matrix and the system is asymptotically stable, the system is stable under angular velocity perturbation.

Stability of the Uniform Perturbed System under Phase-Difference Perturbation.
As concluded in the previous section, the matrix A is a Hurwitz matrix, which is a prerequisite for the stability of the system under phase-difference perturbation. Since Reλ(A) < 0, we can assume Q � Q T > 0, which is satisfying the following Lyapunov equation: e Lyapunov theorem shows that the equation exhibits a unique solution, in which P is a positive Hermite matrix. In this case, the quadratic Lyapunov function V(x) � x T Px must satisfy [26] the following condition: us, the derivative of V(x) along the trajectory of the perturbed system will satisfy where λ min (Q) is the minimum eigenvalue of the matrix Q.
We should note that the first term of right-hand side in equation (34) We can see that the second term Q in equation (35) is a symmetric matrix composed of − (A + A T ). e first term is also a symmetric matrix obtained by the primary row transformation and then further combined similarly and then multiplied by p. erefore, the necessary condition for By substituting Q into equation (32), we have en, we can conclude that the criterion of the symmetric matrix P and Q are positive definite matrixes is 0 < p < 1. Furthermore, substituting the value of the matrix P for the last term of equation (34), we have Mathematical Problems in Engineering where Calculating the determinant values of the matrix G, we have As mentioned above, ρ 2 ≫ 1 and 1 > p > 0; therefore, the matrix G is a positive definite matrix, its eigenvalues must be greater than 0. According to the comparison relation of partial order in matrix theory [28], en, equation (38) can be further converted into the following: As the previous definition x � ε 1 ε 2 T , α � α − α 0 has the same symbols as nx 1 − x 2 because α is an integral mean value point obtained by integrating the perturbation term nε 1 − ε 2 over time in a common period nT 0 , that is, α, x 1 , and x 2 may vary continuously along with the time t; all possible contingencies need to be examined. e first condition, if α > 0, we have x 2 < nx 1 according to equation (43). In this case, equation (42) can be written as For equation (44), W ≤ 0 when x 1 ≥ 0; or W ≥ 0 when x 1 < 0, the system can be stable. e second condition is that when α ≤ 0, there is x 1 ≤ x 2 /n. Correspondingly, equation (42) can be written as Obviously, for equation (45), W ≥ 0 when x 2 ≥ 0; or W ≤ 0 when x 2 < 0, the system is stable.
From the discussions in equations (44) and (45), it is known that if and only if W is equal to 0, the synchronization state of times-frequency vibration system can be made stable under phase difference perturbation. is is certainly true of that W � 0 has solutions. us, the times-frequency system can operate in a steady state.
Actually, the boundaries with large margins are obtained by directly calculating 2x T Pg(t, x) ≤ 0 for such type of zero perturbation system. Even if the perturbation term 2x T Pg(t, x) of the perturbed system is greater than 0, the derivative V(t, x) along the trajectory of the perturbed system can also be calculated by using the general boundary [29]. at is, e origin point can be globally exponentially stable when c < λ min (Q)/2λ max (P). e maximum limit of c can be obtained by taking Q � I in equation (46) [30].
To sum up, the stability of times-frequency vibration synchronization under angular velocity perturbation and phase difference perturbation are proved.

Analysis of the Empirical Phenomena.
According to the mechanical model of the vibration system in Figure 1, an innovative platform is constructed as shown in Figure 3. It mainly includes a times-frequency vibration system, signal acquisition system, high-speed camera system, and some sensors. e structural parameters are uniform, and the relative property is quite strong; two same types of exciters are used, denoted by exciter 1 and exciter 2. Besides, another two exciters with equal mass distributions are also assembled on the frame of the system as balance weight during the test period.
e main parameters of the exciters and the vibration system are listed in Tables 2 and 3, respectively.
To observe the stability of the vibration synchronization state and its phase difference distributions, the different distance l should be considered along with the equivalent radius l e which is a constant for the system. us, l � 0.68 m and l � 1.36 m are taken for the experimental analysis at different rotational speeds of the exciters. A total of 24 groups of experiments are designed. e parameters of the experimental group are shown in Table 4. e experimental process is limited to 100 seconds. To avoid the phenomenon of synchronous vibration transmission, the high-speed exciter turns on firstly and the other exciter motor powers up after 3 seconds delay. If the difference between the maximum value or minimum value of the phase differences of two exciters cannot exceed 30 degrees more than 60 seconds during the experimental process, it will be regarded as achieving the vibration synchronization state, and the average phase differences of steady-state operation will be recorded; otherwise, it will not be considered as achieving the vibration synchronization state. e distributions of phase differences at steady state according to Table 4 are plotted in Figures 4-5, respectively.
In Figures 4 and 5, the left vertical coordinate represents the phase difference, which is statistic data of the steady-state value and represented by boxplot. e size of the box gives intervals of the upper quartile and lower quartile intuitively, in which the straight lines denote the mean value, maximum value, and minimum value, respectively. Also, the (1) (2) (3) (3) (7) (4) (10) Figure 3: Times-frequency vibration synchronization experiment system. (1) Exciter 1, (2) exciter 2, (3) balance weight, (4) rigid frame of the system, (5) four springs with the type of ROSTA AB27, (6) signal acquisition system, (7) two ROLSs (remote optical laser sensors), (8) three triaxial accelerometers, (9) Basler acA1440-220uc camera, and (10) supplement light system for the camera.   corresponding values are also labeled in the figures. e first row of the horizontal coordinate respects the multiple relations of the frequency of the two exciters, and the second row of the horizontal coordinate respects the rotational speed of exciter 1.
Because the phenomena of vibration synchronization transmission appear quickly when n � 1, l � 1.36 m, ω 2 � 1200 r/min, and ω 2 � 1500 r/min, two groups of experimental data are missing in Figure 5. As shown in Figures 4 and 5, the different distribution intervals of phase difference are apparent, the distributions of the boxes in the same frequency and times-frequency experimental groups at l � 0.68 m and l � 1.36 m, the former groups are stable near 180 degrees, the antiphase synchronization phenomenon appears, while the latter groups are stable near 0 degrees and the in-phase synchronization phenomenon comes out. It attributes to the existence of δ in equation (22) leads to the appearance of the bistable phase difference interval. At the same time, the length of boxes denote the vibration synchronization state at n � 1, 2 which is more steady than that when n � 3. e time needed for the same frequency experimental group (n � 1) to achieve the vibration synchronization state is also shorter than other groups both at l � 0.68 m and l � 1.36 m, which can be found in Figure 6. e varying phase differences between the two exciters over time when ω 1 � 1200 r/min, n � 1, 2, 3, and l � 0.68 m and ω 2 � 500 r/min, n � 1, 2, 3, and l � 1.36 m are given in Figure 6 as examples, respectively. e rotational speed of the exciter over time is also plotted in Figure 7. Besides, the experiments also show the rotational speed of exciters has no prominent influence on the stability of the system. From Figure 6, the synchronization state of the same frequency vibration is more comfortable to achieve and more stabilized than that of the times-frequency vibration synchronization state, which is coined with Figures 4 and 5.
e detailed quantitative analysis of the stability of the vibration state can also be found in the next section.
After collecting the pulse signals from the optical laser sensors to calculate the phase differences of the eccentric blocks of exciters, it can be defined as the differences between the angle of rotation φ 1 of exciter 1 to φ 2 of exciter 2 multiplied by the multiple relations of frequency n when the phase of the low-speed exciter is viewed as the reference, i.e., α 2 � 0. At the same time, for an easier way to illustrate the difference of phase differences at l � 0.68 m and l � 1.36 m. e motion of the exciters is referred to a rectangular coordinate system and uses the mass center of blocks of lowspeed exciter passing the coordinate axis in a single circle as a reference. us, the increments of the rotational angle of the exciter and its instantaneous position can be examined over the same time. e corresponding real-time photos of the eccentric block of the exciters obtained by the high-speed camera are shown in Figures 8-13.
In Figure 6 and Figures 8-13, we can quickly note that the phase differences between the two homodromy exciters are stable around 180 degrees when they are located at a short distance; the antiphase synchronization appears. On the contrary, they are around 0 degrees at an in-phase synchronization state, which is coined with Figures 4 and 5 and further verifies the correctness of the theoretical deduction process. At the same time, we note that the distributions of phase differences of the exciters have not the same increase (or decreases) rule. is phenomenon is due to some external unconstant factors, especially the damping coefficients of the axes of the exciters, which are not constants when the exciters operate at different rotational directions and different rotational speeds. Meanwhile, for the vibrating system to achieve the vibration synchronization state, the exciters adjust themselves to the rotational phase continuously to establish the weak interaction.
us, the boxplot does not have the same increase (or decrease) rule.
As mentioned in the previous section, on the one hand, times-frequency vibration machinery urgently needs to be developed from forced synchronization to vibration synchronization. On the other hand, the primary motion type is swing around the mass center of the system for the vibration system driven by two homodromy exciters under the same frequency powered supply when r ψ < � 2 √ as stated in [31], which have no obvious practical significance in engineering. However, the motion of purely swinging around the mass center cannot appear in the system of times-frequency vibration synchronization due to the inconsistent rotational speeds of the two exciters. To observe the trajectories of the system, a feeding material chamber is assembled upon the mass center of the system, and it enriches a part of the particle materials. en, the influence of exciting force and trajectories of the materials can be visually obtained by observing the movement of material. We find that the particle materials in the chamber have not moved as a whole at ω 1 � 1200 r/min, ω 2 � 1200 r/min, and l � 0.68 m and ω 1 � 500 r/min, ω 2 � 500 r/min, and l � 1.36 m. e former phenomenon is attributing to the system operates in a swing state. e latter is at the root of lacking enough exciting force when the two exciters rotate with a low speed. e other conditions denote that the particles are moving with the opposite direction against the exciters' rotation direction, i.e., the particles rotate as a whole in the clockwise direction when the two exciters rotate in the counterclockwise direction. Obviously, the forces acting on the particles come from the vibration of the shell; the exciting loads in the different positions at n � 1, 2, 3, ω 1 � 1200 r/min, and l � 0.68 m and ω 2 � 500 r/min and l � 1.36 m are shown in Figure 14. At the same time, the times-domain responses of them are also plotted, as shown in Figure 15 and Figure 16 As shown in Figure 14, because of the existence of times-frequency loads, the force acting on the shell will not be regular circular, but other forms. Compared with the same frequency loads, the complex compound loads of the shell make the particle trajectories more complicated, which means that there are more mutual extrusion and friction in the particle materials in the chamber along with the process of particles rotating as a whole. It may have apparent practical significance in engineering. For example, for the vibration mill, the group motion of the particle materials will increase the actual contact friction surfaces and the crushing effects of the particles, which may significantly reduce the low-energy zone of milling particles and improve the milling efficiency. e related research is being pushed ahead, which may be discussed in future studies ( Figure 16).

Quantitative Analysis.
As mentioned above, the phenomenon of bistable phase interval in the system is caused by the angle δ when equation (21) is arranged in the form of equation (22). Specifically, W sic (i � 1, 2) is greater than or equal to 0 because of β ∈ [0, π/2]. us, the positive or negative sign of n 2 W c1c + W c2c determines the stability interval of the phase difference of the vibration synchronization system, which mainly varies along with the parameters r l and β. e influence of r l and β on δ is plotted in Figure 17. As shown in Figure 17, the influence of r l and β on δ is more prominent.
ere is only one steady-state interval when β > 30°. e smaller β denotes the more steady phase difference on δ. It just can explain why many vibration machines require the composite mass center of the exciters, and the mass center of the vibration system is in a line to make the steady-state phase difference of the system approach to 0 or 180, in which the synchronous state is more stable. For the experiments in the previous section, we have r l � 1.56 and β � 20.9°when l � 0.68 m and r l � 0.82 and (c) (d) Figure 8: Instantaneous phase of the exciters when l � 0.68 m, n � 1, ω 1 � 1200 r/min, and ω 2 � 1200 r/min: β � 10.8°when l � 1.36 m. Substituting the value of δ and the electrical parameters of exciters into equation (22), the phase difference between the simulation values and experimental results in Figures 4 and 5 are less than ten degrees, respectively, which can also be viewed in Figure 18. Also, the corresponding synchronization performance index D calculated by equation (24) is listed in Figure 19, where the vertical coordinate is logarithmic.
As discussed in Section 1, to obtain a stable synchronization state for a system relying on the adjustment of oscillating objects due to their weak interaction. e more long distance between the two exciters means more weak interaction. is point can be coined in Figure 19 when r l < 2. Actually, r l is usually less than 2 in engineering. us, it is necessary to design the vibration system with a relatively shorter distance between the two exciters for achieving a robustness steady state, especially for the times-frequency vibration system. Compared with two and three times-frequency vibration synchronization system, the same frequency vibration system has more obvious stability. Also, two times-frequency systems is also more stable than three times-frequency synchronous systems, as shown in Figure 19. At the same time, the inflection point of bistable intervals shows poor stability, especially for the system with β � 0.
Besides, for the bistable phase interval approaching to 0 or 180 degrees, the stability criterion (equation (31)) under angular velocity perturbation can be satisfied even if τ 1 and τ 2 are very small when using some other torque-slip models of the mechanical characteristic for the asynchronous motor. Similarly, the stability criterion of the phase difference perturbation system under phase difference perturbation can also be proved by substituting the parameter α and some other required parameters. erefore, under the condition that the stability of the system can be satisfied, the synchronization performance index of the system becomes more significant in engineering.

Conclusions
e mechanical model of a times-frequency vibration system driven by two homodromy exciters is established to investigate whether the phenomenon of bistable phase difference intervals exists in the system. Based on small parameter perturbation methods and nonlinear system theory, the system has been studied with detailed theoretical and experimental research. Some useful conclusions can be given as follows.
Firstly, by introducing a group of average angular velocity perturbation parameter ε 0 and two groups of phasedifference perturbation parameters ε 1 and ε 2 , the frequency capture conditions for times-frequency vibration synchronization of the system are derived. e phenomenon of bistable intervals of the phase difference of the system is revealed. When r l is smaller, we have n 2 W c1c + W c2c ≥ 0; the