1. Introduction

Vibrating screening equipment is essential in engineering for screening mineral materials. They can efficiently categorize different materials based on their shape, viscosity, weight, and other properties. Vibrating screening equipment typically includes a base, a screen box, springs, and driving actuators which are the vibration source. Traditional screening equipment usually relies on gears, chains, and other devices for forced synchronization, enabling the vibration system to promote its amplitude and acquire an effective screening. However, too many connected devices will decrease the overall reliability of the system and cause the economy to decline. Thus, the method of vibration synchronization is proposed to solve this series of problems in engineering [1–4]. The synchronous phenomenon belongs to a branch of nonlinear vibration and was first discovered by Christiaan Huygens in the 17^th century [5–7]. This phenomenon can be observed with two clocks on a pendulum plate under specific conditions, while it vanishes on a stationary wall. Due to the widespread application of synchronous phenomena in engineering, many researchers are dedicated to this field [8–10]. Blekhman and Yaroshevich first studied the self-synchronization theory in 1880 [11]. His research presents the dynamic conditions that enable a vibration system to achieve stable synchronization. Although this theory enables the application of synchronization theory in engineering, the effect of actuators on the vibration system is not taken into account. Zhao et al. not only established the electromechanical coupling dynamics model by considering the driving actuators but also generalized the dynamic model from the plane model to the space model as well [12]. In their work, the problems of synchronous condition and stability criterion are transferred to the problem of solving eigenvalues in characteristic polynomials. Thus, the structure parameter of the vibrating system can be analyzed more intuitively. Fradkov and Nijmeijer studied the problem of synchronization of trajectories in nonlinear oscillatory systems [13]. In their research, they studied the synchronous motion from the perspective of the frequency domain rather than the time domain. The global synchronization with frequency-domain conditions is established based on a filtered strongly oscillatory signal (FSOS) property which bears an external excitation. Zhang et al. conducted a research study on the characteristic analysis using the Runge–Kutta method. The stability criterion of the synchronous states is derived from the Hamilton principle, and the magnitude–frequency characteristics of the vibrating system are discussed [14]. Moreover, the vibrating system with rollers driven by vibrators is also investigated [15]. To obtain more accurate and stable synchronization results, Amador and Brincker proposed a robust and accurate method formulated in the frequency-domain modal model for extracting dynamic properties from vibration data [16]. Three vibratory application examples are given to solve the problem in extracting the dynamic properties from the measured data in an accurate and robust manner.

To achieve an effective screening process in engineering, it is often necessary to realize the elliptical motion trajectory of the vibration system. The self-synchronization motion is limited by its intrinsic dynamic characteristics, so it is essential to maintain sufficient distance between each actuator to realize self-synchronization motion, which makes the structure of the vibration system difficult to achieve miniaturization. This result makes it difficult for vibrating screening equipment to be used in some small structures. With the rapid development of control methods, artificial intelligence control methods are applied to motor and dynamical model control in engineering [17–20]. This is a helpful reference for developing the control synchronization theory. Kong et al. realized the controlled synchronization and composite synchronization by introducing the controlling method of adaptive sliding model in a vibrating system [21, 22]. In their work, the lack of self-synchronization is analyzed and the elliptical motion of the vibrating system is realized with the proposed method. Additionally, many researchers introduced controlling methods into the synchronous phenomenon [23–25].

The synchronous motions mentioned above are based on the same frequency. However, some vibrating machines need different frequencies to satisfy the requirement of mixed materials of varying shapes and stickiness in engineering, such as tungsten screening equipment. Therefore, the method of multifrequency vibration synchronization is presented. Zhang et al. realized double- and triple-frequency synchronization in a mechanical system [26]. Zou et al. obtained the same results with different methods [27]. Although multifrequency self-synchronization motion can be realized, it can only be realized for integer multiple frequencies (n = 2, 3), which limits the diversity of material screening types. Thus, Jia et al. first showed the multifrequency controlled synchronization based on the noninteger parameters of the fixed frequency ratio (n = 1.1–1.9) [28, 29]. In their work, the multifrequency controlled synchronization is realized with two and three motors. As previously stated, it is necessary to save cost in the project, so Jia and Liu introduced the theory of composite synchronization into multifrequency synchronization to realize multifrequency composite synchronization with three motors [30]. Due to its symmetrical design, the vibration system of four motors can achieve stable synchronous motion, providing the advantage of increased amplitude [31].

As the number of motors increases, the electrical equipment required to control them also increases, which results in higher costs. Thereby, aiming to reduce the number of controlled motors, the multifrequency composite synchronization is realized in this paper. According to documents [28–31], the coupling terms of the moment of inertia and stiffness matrix are zero. There is no coupling influence between the ERs. The rotation force of the inductor motors becomes external load disturbances with each other, which makes the system difficult to be stable. Thus, a stable and effective control method should be introduced into the vibrating system. First of all, the dynamical model of multifrequency composite synchronization with four eccentric rotors (ERs) driven by inductor motors is established. The synchronization and stability conditions are given, and the influence of dynamical parameters on the vibration system is described in detail. In addition, combined with the model of induction motors, an improved fuzzy adaptive sliding model (FASM) is proposed. The stability of the controlled system is given by the Lyapunov criterion and LaSalle invariant set theorem. The study further investigates the rationality of the proposed multifrequency composite synchronization theory, as well as the arbitrariness of the nonintegral multifrequency parameter n. Finally, some conclusions are expounded.

2. Methods

2.1. Dynamical Model Establishment

The structure of vibration machines in engineering is extremely complicated as usual. So, to describe the vibrating system better, the dynamical model of the vibrating system is simplified as Figure 1 and the explanations of all the symbols in this article are given in Table 1. A rigid support is fixed to the ground. Four springs which provide the stiffness and damping are used to connect the support to a rectangular rigid test bench. They are symmetrically distributed in four corners. Four motors are symmetrically installed on the test bench along the horizontal axis and the vertical axis, respectively. Each inductor motor is, respectively, installed with a group of ERs, where $$, $$, and l_i = oo_i (i = 1, 2, 3, 4). o is the center of the rectangular rigid test bench and o_i is the center of the ERs. $$ and $$. Through the dynamical model in Figure 1, the mathematical model can be derived by Lagrange equation which is expressed as equation (1):

$$ ()

where T represents the kinetic energy and consists of three items. One is $$ which is caused by the horizontal and vertical movements of the rigid test bench. Another is $$, and it results from rotational movement of the rigid test bench. The other is $$. It originates from the movements of four ERs, where $$, $$, $$, and $$. According to three items above, the expression of T is T = T_teb + T_reb + T_kee. V represents the potential energy of the vibrating system and it is expressed as V = (k_xx² + k_yy² + k_ψψ²)/2. $$ is the item of the dissipation of energy. Finally, the generalized item is divided into two parts, which are, respectively, described as $$ and $$. Therefore, the mathematical expression based on differential equations can be derived as equation (2):

$$ ()

Table 1. The nomenclature table of the symbols.

Symbol	Significance
mi	The mass of each ER
m0	The basic mass of ERs
Jp	The moment of inertia of the rigid frame
m	The mass of the rigid bench
J	The moment of inertia of the vibrating system
kx, ky, kψ	The stiffness coefficients in the x, y, and ψ directions
r	The eccentric radius of the inductor motors
M	The total mass of the vibration system
fx, fy, fψ	The damping coefficients in the x, y, and ψ directions
l1, l2, l3, l4	The distance between the center of the body and the rotating center
f1, f2, f3, f4	The damping coefficients of the ERs
Ji	The moment of inertia of the inductor motor
d-, q-	The d-and q-axes in rotor field–oriented coordinate
Ls	Self-inductance of the stator
Lr	Self-inductance of the rotor
Subscript s/r	Stator/rotor
Lm	Mutual inductance of the stator and rotor
Lks	Leakage inductance of the stator
ϕsd/ϕsq	The flux linkages of the stator in the d-/q-axis
ϕrd/ϕrq	The flux linkages of the rotor in the d-/q-axis
Rs	The stator resistance
Rr	The rotor resistance
Rks	Equivalent resistance of the stator
isd	The current of stator in the d-axis
isq	The current of stator in the q-axis
ird	The current of rotor in the d-axis
irq	The current of rotor in the q-axis
ω	The mechanical speed
ωs	The synchronous electric angular speed
$$, $$, $$, $$	derivation of ϕsd, ϕsq, ϕrd, ϕrq
σ	The leakage factor
Tr	A rotor time constant
usd	The voltage of stator in the d-axis
usq	The voltage of stator in the q-axis
urd	The voltage of rotor in the d-axis
urq	The voltage of rotor in the q-axis
np	The number of pole pairs of the induction motor
ω1, ω2, ω3, ω4	The speeds of four motors
φ1, φ2, φ3, φ4	The phases of four motors
θ	Synchronization electric angle

The parameters of the vibrating system are listed in Table 2. In this section, the items of electromagnetic torque T_e and load torque T_L are mainly discussed. The load torque T_Li can be expressed as equation (3):

$$ ()

Table 2. The parameters of the vibrating system.

Parameters	Values
M (kg)	304
Jp (kg.m2)	44.5
kx (N/m)	129,332
ky (N/m)	105,334
kψ (Nm/rad)	30,715
fx (Ns/m)	615.5
fy (Ns/m)	618
fψ (Nsm/rad)	180.2
θ1, θ2, θ3, θ4 (°)	30, 150, 270, 330
m0 (kg)	6
r (m)	0.05
l1, l2, l3, l4 (m)	0.45, 0.45, 0.45,0.45

The electromagnetic torque is closely related to the current of the inductor motor, so the model of the inductor motor should be firstly discussed. In this article, the inductor motor of squirrel cage is adopted, and its inner structure exists as a short loop which leads to u_rd = u_rq. When the inductor motors operate stably, ϕ_rd = constant and ϕ_rq = 0. There are five fundamental variables in the inductor motor model and three of them (ω-i_s-ϕ_r) are chosen to establish the mathematical model of the inductor motor which can be acquired as equation (4) according to document [31].

$$ ()

where $$, $$, θ = ∫(ω + ω_s)dt, and ω_s = L_mi_sq/ϕ_rdT_r, i_sd = ϕ_rd/L_m. According to the model above, the method of vector control is introduced and the diagram of rotor flux–oriented control (RFOC) is shown in Figure 2. Values with sign “∗” are given values or obtained from the given values. Consequently, the inductor motor can be controlled through i_sq, and the parameters of the four motors are listed in Table 3.

Table 3. The parameters of four motors.

Parameters	Motor 1	Motor 2	Motor 3	Motor 4
P (kW)	0.2	0.2	0.2	0.2
np	3	3	3	3
f0 (Hz)	50	50	50	50
U/V	220	220	220	220
n (r/min)	950	950	950	950
Rs (Ω)	40.5	40.5	40.5	40.5
Rr (Ω)	12	12	12	12
Ls (H)	1.21275	1.21275	1.2175	1.21275
Lr (H)	1.222	1.222	1.225	1.222
Lm (H)	1.116	1.116	1.116	1.116
$$ (Wb)	0.98	0.98	0.98	0.98
f1,2,3,4 (Nms/rad)	0.005	0.005	0.005	0.005

2.2. Stability and Synchronous Conditions

To discuss the theoretical model of equation (1) better, it should be first described in detail. Motor 1 and Motor 4 realize the self-synchronization motion and the result can be expressed as ω₁ − ω₄ = 0 and φ₁ − φ₄ = 0, while the vibrating system is stable. Motors 2 and 3 realize the controlled synchronization motion and its stable state can be converted to ω₂ = ω₃ and φ₂ = φ₃. Motors 1 and 2 realize the multifrequency self-synchronization motion which is described as $$ and nφ₁ − φ₂ = c with the mathematical expression. c represents the constant. p and q are prime numbers with each other, and p/q = n. Averaging the motors’ speeds over their cycles, the four motors’ speeds can be obtained as $$. By introducing the method of the small parameter, it can be set as $$, $$, $$, $$, $$, and $$, where ω₀ is the fundamental frequency and it is a constant. Finally, the three responses in equation (2) from Lines 1 to 3 can be obtained as equation (5):

$$ ()

where tan γ_xi = 2ζ_xω_x/(μ_xiω_i), tan γ_yi = 2ζ_yω_y/(μ_yiω_i), tan γ_ψi = 2ζ_ψω_ψ/(μ_ψiω_i), r_m = m₀/M, $$, $$, $$, $$, $$, $$, r_li = l_i/l_e, η_i = m_i/m₀(i = 1, 2, 3, 4), $$, and $$, $$. The relationship between r_li with r_m and l_i are, respectively, shown in Figures 3(a) and 3(b).

Combining the method of small parameter with the composite synchronous method in document [30], the formulation of Line 4 in equation (2) can be derived as equation (6) because φ₂ = φ₃:

$$ ()

where $$, $$, and $$. The items of T_e0 and k_e0 can be calculated by literature [12]. The parameters in the item k_e0 can be obtained by the nameplate parameters of the inductor motor, but T_e0i changes with some variable parameters. Combining with the electromechanical coupling dynamical model of multifrequency composite synchronization, the feature of T_e0i is illustrated in Figure 4. Taking the small parameters and equation (5) into equation (3) and integrating them over the 2π period, the average load torque $$ can be derived as equation (7):

$$ ()

where the coefficient items of a_ij, b_ij, and κ_i are shown in Appendix A. Assume that the phase difference between Motors 1 and 4 is φ₁ − φ₄ = 2α. Expanding α at point α₀ by the Taylor formula and neglecting the higher order items, α can be expressed as $$ by introducing the small parameter ε₄. $$ is the average perturbation value of ε₄. Then, the coupling equation of the vibrating system can be obtained as equation (8):

$$ ()

where $$, $$, $$, $$, and $$. The items $$, $$, and υ_i are listed in Appendix B. The features of a_ij and b_ij with r_l and 2α are shown in Figure 5. When the system reaches the synchronous state, the small parameters ε = 0 and $$. Then, the synchronous conditions can be expressed as equation (9):

$$ ()

where T_eN1, T_eN2, and T_eN4 are, respectively, the rated electromagnetic torques of each motor. When the synchronous state is stable, υ = 0 and equation (8) can be derived as $$. If Matrix A is with full rank and det |A| does not equal zero, $$ can be represented as $$, where D = A⁻¹B. The characteristic equation can be acquired with |λI − D| = 0 as equation (10):

$$ ()

where d_i(i = 1, 2, 3, 4) represent the coefficients which are given in Appendix C and λ is the eigenvalue. The stability criterion of the composite synchronization motion can be obtained as equation (11) with the Hurwitz condition.

$$ ()

When all the eigenvalues in equation (10) have negative real roots, the synchronous state of four ERs is stable. Otherwise, it is not stable. This result is illustrated in Figure 5(h), and it can be concluded that there is no intersection region of the curves which can satisfy the requirements in equation (11). So, the multifrequency self-synchronization between Motors 1 and 2 cannot be realized, which result in the conclusion in Ref. [31]. To sum up, the multifrequency composite synchronization motion referred in Figure 1 cannot be realized based on the condition of multifrequency self-synchronization between Motors 1 and 2. To realize the multifrequency composite synchronization motion in Figure 1, the multifrequency controlled synchronization motion between Motors 1 and 2 should be realized. Thus, a modified FASM controlling method is proposed.

2.3. The Fuzzy Sliding Model Controller Design

In this section, a master–slave controlling strategy is introduced in the controlling system which is shown in Figure 6. ω_d is the initially given speed of Motors 1 and 4. φ₁ and φ₂ are, respectively, the target phase and actual phase in the multifrequency controlled synchronization between Motors 1 and 2. Similarly, φ₂ and φ₃ are, respectively, the target phase and actual phase in the same frequency controlled synchronization between Motors 2 and 3. According to the master–slave controlling strategy above, a FASM method based on the controlled input (CI-FASM) is proposed. Thus, the target phase can be uniformly expressed as φ_d. In the meanwhile, φ is given as the actual phase. The structure diagram of CI-FASM is shown in Figure 7. First, the integral sliding mode surface is defined as equation (12):

$$ ()

where e is the tracking error, and e = φ − φ_d. k₁ and k₂ are nonzero positive constants. When the sliding mode control is in an ideal state, $$, which can be converted to $$. Only if the parameters k₁ and k₂ are appropriately adjusted, the tracking error will approach zero. Second, rewriting Line 4 in equation (2) combined with equation (4), equation (13) can be derived as follows:

$$ ()

where Φ = 3L_mϕ_rd/2L_r, U = i_sq. The control law can be obtained as equation (14):

$$ ()

where g(φ) = Φ/J, $$, and Γ = −T_L/J. Because the multifrequency self-synchronization between Motors 1 and 2 cannot be realized, the ideal control law is hard to be satisfied. Then, a method of fuzzy approximation is introduced to solve this problem. Set σ_i as a group of adjustable parameters and ξ_i as a group of fuzzy basis vectors. The control law is derived as u_fz(s, σ) = σ^Tξ^T, where $$, ξ = [ξ₁, ξ₂, ⋯, ξ_m], and $$. Through searching an optimization u^∗ = u_fz(s, σ^∗) of the fuzzy system to approximate the ideal control law, it can be described as equation (15):

$$ ()

where ρ is the approximate error and |ρ| ≤ E. E is the switching gain. Replacing σ^∗ with the estimated value $$, the control law is expressed as $$. Finally, introducing the switching control law u_vs to compensate for the error between u_fz and u^∗, the total control law is u = u_fz + u_vs.

3. Results

In this section, some numerical simulation examples are given to certify the correctness of the multifrequency composite synchronous theory first. Then, the effectiveness and stability of the controlling method and the arbitrariness of the multifrequency parameter n are both illustrated. The experiment is shown to demonstrate the consistency among the three parts. Finally, the novelty and research contribution are with an in-depth discussion.

3.1. Numerical Simulation Results

A multifrequency synchronization method is proposed to provide complex trajectories and avoid the issues of blocking and sticking in vibrating screens used in engineering. However, the multifrequency self-synchronization method is difficult to apply to the integrated harvesting and screening machine due to the structure size which is affected by the instinct dynamical feature of the vibrating system. To accommodate the installed requirement, the structure size is reduced with the CI-FASM controlling method. Aiming at the four ERs, fully controlled synchronization incurs significantly higher costs. Therefore, the multifrequency composite synchronization method, which integrates multifrequency controlled synchronization with self-synchronization, is proposed to reduce expenses. The numerical simulation results are shown below. According to the approach of the structure in Figure 6, the initial target speed in Figure 8 is given as 60 rad/s and the multifrequency parameter is shown as n = 1.5. From Figure 8(a), it can be known that the speeds of Motors 1 and 4 are stable at the given speed. The speeds of Motors 3 and 4 both reach 90 rad/s with little floating. This result proves that the synchronous speeds can be realized with ω₁ − ω₄ = 0, ω₂ − ω₃ = 0, and 1.5ω₁ − ω₂ = 0, which is proposed in the “Methods” section. In Figure 8(b), the three groups of phase differences all approach zero, which can be expressed as φ₁ − φ₄ = 0, φ₂ − φ₃ = 0, and 1.5φ₁ − φ₂ = 0. This result indicates that Motors 1 and 4 realize the self-synchronization motion, Motors 2 and 3 realize the controlled synchronization, and Motors 1 and 2 realize the multifrequency controlled synchronization. Thus, the vibrating system realizes the multifrequency composite synchronization. Moreover, compared with the former literature, the proposed CI-FASM controlling method presents better controlling effectiveness which contains smaller tracking errors of the phase and a faster controlling response. Figure 8(c) shows the load torques of four motors which are all between −1.3 and 1.3. This result illuminates that the load torques are lower than the electromagnetic torques which can be calculated with the parameters in the three tables. Thus, the four motors can operate normally and the phenomenon of locked rotor will not occur. Figures 8(d) and 8(e) demonstrate the responses of three directions. Because the two groups of motors both operate in the opposite direction, all the forces in the x direction cancel out with each other. Thus, the response in the x direction is approximately 0, and this result is consistent with the dynamical feature in Figure 1. The response in the y direction is composed of a superposition of amplitudes with two different frequencies. The response in the ψ direction fluctuates around 0°. This result indicates that the rigid body operates with a very little swing angle, and the vibrating system can reach the stable multifrequency composite synchronization which is proposed in the theory part.

In this numerical simulation example, the multifrequency parameter n is adjusted from 1.5 to 1.2 for certifying its arbitrariness. Figures 9(a) and 9(b) show the speeds and the phase differences of four motors. In Figure 9(a), it provides the result of 1.2ω₁ − ω₂ = 0, while the result of 1.2φ₁ − φ₂ = 0 is shown in Figure 9(b). Figure 9(c) reflects the influence of the different multifrequency parameter n. Compared with Figure 8(c), this result indicates that the amplitude of load torque becomes smaller as the parameter n decreases. On the contrary, the amplitude of load torque will become larger as the parameter n increases. Figure 9(d) gives a different response in the y direction. And this result shows the diversity of the motion trajectory of the vibrating system with different multifrequency parameters n. In Figure 9(e), the swing angle ψ still approaches 0, which confirms that the vibrating system reaches the stable multifrequency composite synchronization state. In the meanwhile, the stability and effectiveness of the controlling method and the arbitrariness of parameter n are also proved.

3.2. Experiment Validation

To certify the correctness of the theory proposed, an experiment of the multifrequency composite synchronization is designed as Figure 10. The vibrating testing bench (Figure 10(e)) is installed on the ground. Four inductor motors are, respectively, controlled by four converters (Figure 10(f)) which are powered by the three-phase alternative current supply. Four hall magnetic switches (Figure 10(b)) are, respectively, fixed on the vibrating testing bench corresponding to four motors. Each ER of the four motors is, respectively, stuck with a magnetic patch. When the motor turns one circle, a pulse is transmitted to the INV (Figure 10(a)) which connects with the PC (Figure 10(c)). Four pulse encoders (Figure 10(g)) are installed on a steel frame, and each pulse encoder connects the corresponding motor shaft with a rubber hose. The other side of the pulse encoder is connected to the PLC (Figure 10(h)). Then, the PLC connects to the converter. The air switch (Figure 10(i)) converts the 380 V AC to the 24 V DC and provides it to the PLC. The three-axial acceleration sensor (Figure 10(d)) is fixed on the vibrating testing bench to measure the response amplitude which is transmitted to the INV.

As shown in Figure 11, the experiment with the parameter n = 1.5 is finished to illuminate the correctness of the theory. In the experiment, the power supply frequency of Motors 1 and 4 are set as 27 Hz. The power supply frequency of Motors 2 and 3 are set as 40.5 Hz. Then, the speeds of the four motors can be calculated according to the pole pairs of inductor motors which are given in Table 3. According to the formulation ω = 2πf/n_p, the speeds of Motors 1 and 4 can be calculated as 56.52 rad/s. Similarly, the speeds of Motors 2 and 3 are 84.78 rad/s. From Figure 11(a), the speeds of Motors 1 and 4 both fluctuate around 56.46 rad/s. The speed error is about (56.52 − 56.46)/56.52 ≈ 1000. It can be recognized that Motors 1 and 4 both realize the synchronous speed. Similarly, the speeds of Motors 2 and 3 fluctuate around 84.5 rad/s and the error is about 3000%. Figures 11(b), 11(c), 11(d) present the phase differences of four motors. The absolute phase difference between Motors 1 and 4 is about 6°. Compared with the theory φ₁ − φ₄ = 0, this result can be deemed that Motors 1 and 4 realize self-synchronization in engineering. The absolute phase difference between Motors 2 and 3 fluctuates around 5°. Thus, Motors 2 and 3 realize the controlled synchronization with zero phase difference, which verifies the theory assumption φ₂ = φ₃ in the model of multifrequency composite synchronization. The absolute phase difference between Motors 1 and 2 is about 13°, which is consistent with the theory nφ₁ − φ₂ = c. Figures 11(e), 11(f), 11(g) show the responses of the vibrating system. The response in the x direction is much smaller than it is in the y direction. This result is consistent with the theory in Equation (2) and the numerical simulation shown in Figure 8(d). Moreover, the curves in Figures 11(f) and 11(g) have the same tendency as the curves in Figure 8(d). According to the results above, the multifrequency composite synchronization can be realized with proposed CI-FASM controlling method.

4. Discussion

Combined with the numerical simulation and experiment results, the multifrequency composite synchronization method can be implemented. In the meanwhile, it is possible to screen different kinds of materials by adjusting the fixed frequency ratio parameter n. It is proved that the proposed method can be applied to multiple frequency vibrating screens in engineering.

The mainstay of the proposed method is first to confirm whether the miniaturized dynamic model presented in this paper can be achieved through the multifrequency self-synchronization method. If the multifrequency self-synchronization is suitable for the miniaturized vibrating screen system, the control method proposed in this paper becomes irrelevant. According to documents [26, 27], the synchronization becomes better with decreasing r_l. When η_i = 1, the mass of four ERs equal with each other and r_l < 1.09. Conversely, when η_i ≠ 1, the synchronism of the vibrating system is worse. This result is also reflected in Figure 5. This result indicates that when η_i = 1, r_l has the minimum value, the vibrating system will acquire the best self-synchronization condition. Thereby, the condition η_i = 1 is applied in the synchronization analysis. According to the Hurwitz condition in Figure 5(h), although parameters suitable for multifrequency self-synchronization are used, the multifrequency self-synchronization based on the dynamic model described in this paper still cannot be achieved. Then, it is necessary to apply control methods to the dynamical model. In addition, according to documents [21, 22], synchronization with the same frequency that aims to enlarge the amplitude of the vibrating system can usually only achieve the screening of a single material. The screening effect of mixed materials is not ideal in engineering.

By introducing the master–slave control strategy and the CI-FASM control algorithm, the multifrequency composite synchronization and the arbitrariness of the parameter n are realized. Compared Figure 8(d) with Figure 9(d), different parameter n shows different responses. This result indicates that the method proposed in this paper can give a suitable fixed frequency ratio for various or mixed materials to obtain a better screening process. Although the document [31] solves the problems above, there are many electrical equipment, which lead to high costs. Then, the control method in document [31] is hard to satisfy the stability of the dynamical model in this paper. Compared with the former literature [21, 22, 28–31], the proposed CI-FASM method presents better control effectiveness as well as smaller tracking phase errors. The multifrequency composite synchronization method is used to reduce the electrical equipment and finally reduce the cost.

During the experiment, there may be a resonance phenomenon between the motors and the vibration system, which can lead to a blocking phenomenon of the motors. The motors’ frequencies are so low that the energy is not enough, and the jump phenomenon of near resonance (Sommerfeld effect) will appear. This is not beneficial for the controlling result. So, the motors’ frequencies should be well chosen and suitable for the converters in Figure 10(f). Besides that, the structural vibration characteristics of the vibration system are also not taken into account. The modal response of multiple frequencies to vibration systems is not fully considered. We will consider and investigate this feature in our future study.

5. Conclusions

Conflicts of Interest

The authors declare no conflicts of interest.

Author Contributions

Lei Jia wrote the article, Qingsong Chang compiled the figures and tables, Yang Tian led the experiments, Xin Zhang checked the article, and Ziliang Liu conducted the simulation.

Funding

The author’s research was supported by 2022 Liaoning Education Department General Project (Project No. LJKMZ20220602).