Investigation on the Self-synchronization of Dual Steady States for a Vibrating System with Four Unbalanced Rotors

: In the field of vibration utilization engineering, to achieve the maximum degree or the highest efficiency use of the excitation force is still a hotspot among researchers. Based on this, this paper has carried out a series of synchronous theoretical analysis on the four identical unbalanced rotors (IURs) symmetrically and circularly mounted on a rigid frame (RF) model, which is used to drive a cone crusher. The dimensionless coupling equations of the four IURs are established using the improved small parameter method. The analysis of the coupling dynamics characteristics of the system shows that the four motors of the system adjust the speed through the synchronous torque to achieve synchronization, and a parameter determination method for realizing offset self-synchronization to eccentric force was put forward under the steady state of ultra-resonance. Furthermore, the relationship between the equivalent stiffness of the crushed material and crushing force and compression coefficient is discussed, and the design method of the full-load crusher working under the steady state of sub-resonance is proposed. Finally, through a series of computer simulations, the correctness of the self-synchronization of dual steady states is verified.

one for the case that ratio of the distance between the IURs and the mass center to the equivalent rotating radius of the RF is greater than 2 , and swinging motion about its mass center for that less than 2 [11]. But when the two IURs rotate in opposite directions, the motion of the RF is linear one [12]. When the rotational axe of the two IURs are interspace lines and symmetrical about the symmetry axis of the RF, the trajectory of point on the RF is screw line [7,13]. When the rotational axe of the two URs are not symmetrical about the symmetry axis of the RF, the motion of the RF is translational one and swing of plane, and the trajectories of points on the RF are differential ellipses [14]. When two identical auxiliary rigid frames (ARFs) are symmetrically installed on a RF and two identical URs in the same direction are installed symmetrically on each of ARFs, the four IURs can excite the elliptical motion of the RF [10]. But when four IURs are installed on the same RF, exciting forces of the URs can be cancelled mutually and the RF is static under the condition of self-synchronization [15,16]. When three URs are in linear distribution on a RF, the three URs can achieve the RF's swinging and elliptical motion or linear motion by changing three URs' different installation methods [17][18][19][20], but the operation efficiency of vibrating system cannot be improved in the ultra-resonant state [19,20]. Besides, the anti-resonance system with three URs and two RFs is proposed to explore its motion selection [17,18,21]. It shows that the main rigid frame (MRF) can achieve swinging and elliptical motion while the accessorial rigid frame (ARF) is static by installing two URs on the same axis and symmetrically placing another UR on the other sides [17].
The models of double or multiple URs' self-synchronization have been applied increasingly to field of engineering machinery. Besides vibrating screens, vibrating feeders and vibrating conveyors [7][8][9]22], a VM with two IURs located on two ARFs is proposed to achieve rock crushing in a crusher [23]. Shokhin et al discussed the effect of system parameters on crushing performances in [24].
In this paper, a single RF vibrating system with four IURs is proposed and applied to a cone crusher.
This system operates two steady states of synchronization. One of synchronous states is ultra-resonance for idle state of the crusher, in which the exciting forces of the four IURs are cancelled mutually and the RF is static; another is sub-resonance for material crushing state, in which the exciting forces are supermposed mutually and excite the circular motion of the RF. In the next section, coupling dynamics analysis is done to derive dimensionless coupling equations of the four URs and obtain criteria of synchronization and stability for the four URs. The dynamic parameters of the VM that satisfy the two synchronous states are determined in section 4, and effect of crushed material property on the effective crushing force is discussed in section 5. In Section 6, computer simulation and experiments are conducted to verify the correctness of the theoretical research. Finally, conclusions are given in Section 7. Figure 1 shows mechanism of a material crusher, which consists of a moving cone, a fixed cone, four supportors, and four driving systems. By means of a mandral, the fixed cone is supported on a rigid base on an elastic foundation. One driving system is composed of an asynchronous motor, a belt, a cardan shaft, a supporting shaft and a UR. One supportor is a connecting rod with two spherical linages at its ends. The moving cone is connected with the base by means of four supportors and driven by four symmetrically distributed IURs rotating with four asynchronous motors, respectively. When the system is running, the superimposed exciting forces stemming from the four URs cause the moving cone a circular motion on the horizontal plane, which periodically compresses the V-shaped space between the moving cone and the fixed cone to achieves the purpose of crushing materials.

Motion equations of system
The overall structure of the vibration system is shown in Fig. 2 (a). At the radius L of the moving To facilitate the calculation of the formula, we numbered the four IURs 1-4, among which the face to face mounting UR 1 and UR 2 are divided as a group, so do 3 and 4, as is shown in Fig. 2 (b).
The system has three degrees of freedom, namely the movements in the x and y directions and the swing motion of  around the center of mass G. The four IURs respectively rotate around their own motor shafts, and the angle of rotation is represented by i  , 1, 2,3, 4 i  .
In the coordinate system as is shown in Fig. 2 (c): oxy is the fixed coordinate system of the crusher; Gx y   is the translational coordinate system and its -x axis and -y axis is parallel to thex axis andy axis respectively; Gx y   is the follow-up coordinate system. The coordinates of the eccentric rotor of each UR (hereinafter referred to as an exciter) can be expressed as in Gx y   : cos cos sin sin where R is the installation radius of the exciter center; r is the eccentricity of the exciter with 11 12 21 22 31 32 41 42 r r r r r r r r r         ; i  is the angle between the center of the motor shaft and the x axis with 1 0   , 2 π   , 3 π 2   , 4 3 π 2   .
Moving cone Using coordinate transformation, the coordinate of the exciters in the system Gx y   is transformed into the system Gx y   : with the transformation matrix cos sin sin cos The coordinates in the fixed system oxy can be expressed as: The coordinates of the joints between the springs and the RF are: According to the above coordinates and coordinate transformation, the kinetic energy and the potential energy of the system are respectively: where m is the mass of outer cone; w J is the moment of inertia of outer cone; 0i j is the moment of inertia of motor i; i m is the mass of each eccentric rotor with The energy dissipation of the system can be expressed as a function: where i F is the damping matrix of spring i, 1 According to Lagrangian equation: In the vibrating system, we choose        1  2  3  4 , , , , , , as the generalized coordinate of the system, under each generalized coordinate the generalized forces of the system can be expressed as Substitute equations (5)-(7) into equation (8), in the calculation process, considering that the mass of each exciter of the exciter is much smaller than the mass of the vibrating rigid body, and the inertial coupling caused by the asymmetry of the installation of the exciter is ignored. And according to the literature [6,7] reviewed, the rotation of the vibrating rigid body belongs to a small amplitude rotation, and the rotation angle satisfies 1   , the differential equations of motion of the system are as follows: is the shaft damping constant of motor i; ei T is the output electromagnetic torque of motor i; e l is the equivalent radius of gyration of the system; ) (  , ) (  are the derivatives with respect to time t.

Synchronization of the four IURs system
Assume that the average phase between the four exciters is  , the phase difference between the exciter 1 and the exciter 2 is 1 2 , the phase difference between the exciter 3 and the exciter 4 is 2 2 , the phase difference between the exciters 1＆2 and the exciters 3＆4 is 3 2 , which satisfies: Assuming the average value of the average angular velocity of the four exciters is m  , the instantaneous fluctuation coefficient of the average angular velocity  & is 0  , the instantaneous then the small parameters are set as: Combining the above formula, the phase, instantaneous angular velocity and instantaneous angular acceleration of the four exciters can be expressed as: is the instantaneous fluctuation coefficient of the angular velocity of each exciter for respectively. If the average value i  in a single positive period satisfies 0 i   , then the running speeds of the four exciters are the same, that is, the operation is synchronized. When the vibration system is in a stable state of synchronous operation, the i  & & term in the differential equations of motion in the x, y, and  directions, that is, the effect of angular velocity on the steady state excitation can be ignored [6,7]. Therefore, the equation in the steady state of the system can be written as: Considering the damping coefficient of the spring, The fluctuation coefficient of the speed of each motor i  is too small to affect the excitation response in all directions. Therefore, according to the superposition principle of the linear system, the response in each direction can be approximately expressed as the following form: cos( ) Substitute them into the equation of the unbalanced rotor of the differential equation of motion (9), Integrate  on 0-2π and divide by 2π to get the average value in a single period, after ignoring the higher-order terms of i  and i &, we obtain: .
The ij   , ij  , fi  , ai  in equation (16) are given in the Appendix A. It is worth noting that in equation (16), compared with the changes of the angle of rotation, the changes of 1  , 2  , 3  to time are small, so ( 1, 2,3, 4) i i   can be regarded as a slowly varying parameter. Therefore, in the above integration process, according to the method of direct separation of motion [1][2][3][4][5], we replace In the equation (16), when the rotation speed of the induction motor is in a steady-state operating state near, the electromagnetic torque of each motor can be expressed as: When the four motors are powered by the same source and have the same number of pole pairs, in the stator voltage synchronous coordinate system, they can be expressed as: where si R , ri R represents the stator resistance and the rotor resistance; si L , ri L , mi L represents the stator inductance, the rotor inductance and the mutual inductance; p n represents the number of where A is the dimensionless inertia coupling matrix, B is the angular velocity dimensionless stiffness coupling matrix, u is the dimensionless load coupling.

Synchronization conditions
If the four vibration exciters of the vibration system of the vibrating crusher realize synchronous In the matrix form of the dimensionless coupling equation, we (20) can be simplified to  u 0 , and let: o1 T -o4 T represent the output electromagnetic torques of four motors during synchronization, the vibration system transfers electromagnetic torque between the four exciters by adjusting the phase difference between the four exciters to balance the difference in output torque between the four motors. Introduce the output torque difference: denotes the kinetic energy of the exciter with an eccentric rotor. According to equations (25)-(27), we have: ( , , )     shows the dimensionless coupling torque between exciters 1 and 2, exciters 3 and 4, and exciters 12 and 34 respectively. The left side of equations (29)-(31) represents the difference of the dimensionless residual electromagnetic torques.
The dimensionless coupling torques are function of 1  , 2  , 3  , which satisfies: Among them, c12max  , c34max  , cmax  respectively represents the maximum value of three groups of non-dimensional coupling moments. In order to ensure the synchronous operation of the four exciters, it is necessary to ensure that the above equation has a solution. Therefore, the synchronization criterion for the synchronous operation of the four exciters is transformed into the existence of the solution of the phase difference equation between the exciters. The synchronization criterions are as follows: Define the maximum value of the dimensionless coupling torques as the synchronous torques or the capture torques between the groups of exciters, the meaning of the synchronization criterions is: the synchronous torques between the exciters are not less than the absolute value of the difference between the corresponding dimensionless residual electromagnetic torques.

Stability of synchronization and stable criteria
When  u 0 , the above system is the generalized system [25]:    . When the system is under no-load, it is operating in the state of far beyond resonance, the phase lag angle in all directions is close to π, s0 W , sc W can be ignored, 0 A is symmetrical and 0 B is anti-symmetric, when the parameters of the vibrating system satisfy the follow condition:  T  T  0  0  m0  11  22  33  44 diag , , , where I is the unit matrix.
If satisfies lim t  Aε 0 , then the above-mentioned generalized system is permissible and impulsive-free, and the generalized system is in a stable state, so when lim t  ε 0 , it means that the electromagnetic torque of the four motors and the load torque applied by the vibration system are balanced. Linearizing equations (21)  (24) around 10  , 20  , 30  , m0  , and neglecting s0 W , Summing up equations (39)-(42), we obtain: Substituting equation (43)  , , are specifically given in Appendix B.
Generalized system (44) can be rewritten as: , the characteristic equation of C can be obtained: Only when all characteristic roots have negative real parts, the zero-valued solutions of (45) are stable. According to the Routh-Hurwitz criterion, the coefficients of the above characteristic equation need to meet the following conditions: In the system of this paper, the parameters of four induction motors are selected to be similar: In this case, the matrix E can be simplified to: After multiplying the inverse matrix of E and the matrix D , C it can be written as follows :   11  e0  12  e0  13  e0   21  e0  22  e0  23  e0  3 3   31  e0  32  e0  33  e0   2 The coefficients of the characteristic equation (46) satisfy the following relationship: The system must meet the Routh-Hurwitz criterion after satisfying the Lyapunov criterion. 1 2 cc The subsequent numerical analysis results can be verified at this point. According to the Lyapunov stability criterion and the value of the cosine coupling coefficient cc W , the value range of steady-state phase differences 1 2 , 2 2 can be given as (52).

Numerical analysis
In order to prove the self-synchronization ability and stability of the vibrating system for idle state, it is necessary to analyze the system in the related synchronization problems and the stable state numerically. 0. 05

Steady-state phase differences for idle state
Substitute the above data into equations (21)  (24) and obtain nonlinear equations related to the given by system parameters. Solving them by numerical method, we can obtain the stable motor speed m0  and the phase differences 10  , 20  , 30  when the system operates synchronization, the solutions of the phase difference are: 10 20 30 2 0, π, 2 0, π, 2 0, π 2, π.
In order to reflect the influence of the dimensionless parameter e r on the synchronization and stability of the system more intuitively, it is necessary to discuss the maximum value of it: . So according to the given mass ratio m 0.04 r  , we can get the emax r value of 2.5, that is, in the analysis of synchronization and stability, the value of e r should be limited to between 0 and its maximum value of 2.5.
Combining the Lyapunov stability criterion (36), the Routh-Hurwitz stability criterion (47) and the value range of e r to determine the phase difference combination in (53), we can obtain the relationship between the dimensionless parameter e r and the steady-state phase differences as shown in Fig. 3. The synchronous speed of the system under steady-state phase differences can be calculated through the solution process above, is at the stable value of m0  =104.2 rad/s.

Stability curve
Applying the above numerical analysis to the characteristic equation of matrix C , we obtain the three characteristic roots of C . Take the real part of the three characteristic roots as 1  , 2  , 3  and regard them as the first stability coefficients of the system. By plotting the first stability coefficients' curve, the general trend of stability with the change of e r can be obtained. It can be seen from Fig. 4 that when the dimensional parameter e r of the system change from minimum to maximum, the coefficients 1  , 2  , 3  are always less than 0. 1  maintains at around -2.45,  Fig. 5 shows all the second stability coefficients. Fig. 5 (b) shows that the dimensionless parameter 12  is greater than 0 when e r is greater than 1.414, which verifies the accuracy of numerical calculation of the steady-state phase differences.

Synchronous capability curve
The greater the coupling torques of each group of exciters, the greater the torque differences between the drive motors are allowed when the system implements synchronization. Introduce the synchronous ability coefficients to indicate the ability of the exciters to achieve synchronization.
The right side of each equation in (55) represents the load torque imposed by the vibrating system on each exciter. The first term represents the load torque excited by its own eccentric rotor, and the other terms include the phase difference sine term coupling torques and phase difference cosine term coupling torques between this exciter and other three exciters. It is found that the coupling torques of the sine term between any two exciters has the same sign of action on a single motor, so the system uses the coupling torques of the cosine term between the exciters to limit the increase of the speeds of the phase-leading eccentric rotors and to drive the phase-lagging eccentric rotors. The coupling torques between two exciters can be expressed by the product of the maximum value of the corresponding cosine coupling term and the kinetic energy su T .
When cosine coupling terms are not equal to 0 and The maximum value of (58) is , then the synchronous ability coefficient between the exciter 1＆2 and 3＆4  can be expressed as equation (59).
Through numerical analysis, the relationship between the synchronous ability coefficients and the dimensionless parameters e r can be obtained as shown in Fig. 6. Considering the phase differences stable interval, good synchronous ability and stability comprehensively, we obtain the parameter determination method of eccentric force offsetting self-synchronization for idle state. In this paper, it can be described as the given value interval (1.414, 2.5) of e r . This design can ensure the crusher's actual working conditions, in which the system should be in a self-synchronizing cancellation state and the rigid body, that is, the outer cone should be in the stationary or the slightly vibrating state.

The relationship between crushing force and equivalent stiffness for material crushing state
After the crusher is filled with materials and enters the full-load state, the equivalent stiffness of the system in all directions is based on the original rubber spring stiffness plus the material's equivalent stiffness. The increase of the equivalent stiffness in each direction also increases the natural frequency. While the operating frequency of the motor remains unchanged, the frequency ratio of the system reduces, so that the system transitions from the zone of far exceeding resonance working to the zone of sub-resonant. Under the synchronization condition in the sub-resonance zone, the frequency ratio of each direction is under 1. Solving nonlinear differential equations (21)  (24) through vibrating parameters to obtain phase differences at full load, 10 2 0   , 20 2 0   , 30 2 0   is obtained. The response of the outer cone of the system in three directions of x, y,  is superimposed according to equation (15).
The outer cone and inner cone of the crusher are in contact with the materials through the inner wall and there is a great friction between the inner wall and the materials. Considering that the response of the rigid body in  direction is small, so the friction torque between the inner wall and the materials can offset the response in  direction. The system motion form is a circular motion with the response superimposed in x and y directions. Fig. 7 shows the dynamic model of the plane motion of the outer cone in x direction when it is fully loaded. The dynamic model in y direction is equivalent to that in x direction. When the crusher is full of materials, the system runs synchronously with four eccentric rotors in zero phases, which belongs to the self-driving vibration. The compressed materials can be regarded as a spring, and its equivalent damping is mx f ; Suppose the statistical characteristics of the compressed materials is isotropic, and its equivalent stiffness is the same in all directions, we set its extrusion stiffness to  times the total weight of the moving rigid body, that is: where  is the stiffness coefficient of the materials. The equivalent stiffness and equivalent damping of linear vibration are respectively: m m , .
x rx x

Working amplitude and crushing force at full load
The differential equation of motion of the four-vibrator self-driving vibration in the x direction can be expressed as: that is: From this, the responses and response speeds of the full-load circular motion of the system can be   .

Critical stiffness of crushing materials
The cone crusher adopts two support schemes, which are supported by rubber springs and plane motion bearings between the outer cone and the machine base. In the static state, the rubber springs and plane motion bearings jointly bear the total weight of the outer cone rigid body Mg . In the working process, the extrusion force between the outer cone and the materials produces a downward component, and this part of the force is borne by the plane support bearings. Suppose the proportion of the plane bearings carry the weight Mg is  , 0< <1, the rest is carried by the rubber support springs. Suppose the static deformation of the rubber springs in the vertical direction of vibration is z  , their vertical stiffness is: Suppose the ratio of the vertical stiffness of the rubber springs to the shear stiffness is  ,then the horizontal stiffness is: (1 ) .
where c  is called the critical equivalent stiffness coefficient of the crusher's crushing materials.  In order to explore the influence of the material stiffness coefficient and system design parameters on the frequency ratio of the operating point, the stiffness ratio of the rubber springs  are taken as 25 and 40 respectively, and the bearing support coefficient  are taken as 0 and 0.8 respectively.
The corresponding change rule is shown in Fig. 9. It can be seen from the comparison of Fig. 9 (a) (d)  that the change of  and  of the crusher has little effect on the frequency ratio of the operating point, especially when the operating point frequency is relatively small. In Fig. 9, the frequency ratio contour of the working point changes the fastest in the direction of the change of the material equivalent stiffness coefficient, indicating that the frequency of the working point mainly depends on the equivalent stiffness coefficient of the material.
According to the analysis of the parameter value interval in Fig. 10, the change speed and amplitude of the operating point frequency ratio m0 nx   in the vertical deformation z  direction of the spring are smaller than the change speed and amplitude in the direction of the material equivalent stiffness. Combining the data of spring deformation z  =1.5 mm in Table 1 and Fig. 10, it can be known that the system must work in the sub-resonance zone to ensure that the operating frequency of the system is less than the natural frequency under full-load condition, and the value of the material equivalent stiffness coefficient must not less than 1100, which is numerically consistent with the material critical equivalent stiffness coefficient in Fig. 8.

Material effective crushing coefficient and discharge port compression coefficient
For the vibrating system shown in Fig. 7, the critical damping is: Set the damping coefficient of the spring and material in x direction as: Substituting (76) and (77) The forces between the outer cone and the material are: Then the magnitude of the restoring force of circular motion is: The magnitude of the force between the outer cone and the materials is: The ratio of restoring force to the excitation is defined as the system restoring force coefficient s  : The ratio of the force between the outer cone and the materials to the excitation is defined as the crushing force coefficient m  : The ratio s  to m  is defined as the effective crushing coefficient of the crusher. Higher crushing efficiency and better crushing effect needs larger effective crushing coefficient.
According to equation (64), the radial displacement of the outer cone during the working process can be calculated, and the compression coefficient of the discharge port can be obtained by the ratio of the circular radial movement to the nominal amplitude as: Discuss the material equivalent stiffness coefficient  and the system damping ratio x  to the above coefficients. Fig. 11 and Fig. 12  From equation (87), we can see that  ,  , z  are much less than  . Also it can be seen from The V-shaped space between the outer cone and the fixed cone periodically changes in amplitude and the amplitude decreases. At this time, the smaller and finer crushed particles will be obtained. It is necessary to adjust the material equivalent stiffness coefficient  to adjust the actual working frequency, which is at the sub-resonance zone critical point, to meet the crushing requirements according to specific engineering practice and crushing particle size requirements.

Computer simulation
In order to prove the correctness of the above theoretical research results, the computer simulation method is used for vibrating analysis. The digital simulation is based on the principle of numerical calculation. In this paper, we use Kutta-Runge theory and C language to compile a simulation program to analyze the synchronous speeds, steady-state phase differences, and displacement responses in each direction of the plane motion.

Idle state analysis
The The computer simulation analysis results of the four-machine self-synchronous speeds are shown in Fig. 13. Since the four motors are supplied the same frequency power, so the four exciters driven by the four motors are also the same. The steady-state speeds of the four motors tend to be the same, and the accelerations are also equal. As shown in Fig. 13, the speeds of the four motors began to converge at 4s, and were in a fluctuating state before then. After 5s, the synchronous torque between the four motors makes the speeds of the four motors start to be in a stable state. The final synchronous operating speed is at around 998.5 r/min. have reached the vibrating synchronous transmission state, that is, the natural frequency of the system is increased by adding the material stiffness to adjust the operating frequency point of the system. The speeds of the four motors will reach the steady state again in a short period of time, and the system will be stable for the second time. The state of the body changes the motion form of the body through the change of the phase differences.

Full-load simulation analysis
Corresponding to the idle state, the material crushing state is another critical state of the crusher system. The phase differences and displacements of the critical state after the crusher is full of materials are specifically analyzed.  Fig. 13, which is maintained at around 104.2 rad/s. At this time, the phases between the four motors all tend to be the same to make the linear responses superimpose. The excitation responses are shown in Fig. 16. The amplitudes in the x direction and y direction are approximately equal at around 3.5 mm. In the  direction, because the rotation damping is large and the value of the rotation stiffness is small, the rotation angle gradually approaches 0 with time. In this state, the crusher realizes the selection of elliptical motion.

Experimental tests and verifications
To examine the correctness of our theory, numerical and simulation results, we carried out some experimental tests and verifications in this section. The point that the real body of the vibrating cone crusher is hard to be manufactured and fabricated, because of its enormous size (more than 3 meters in length), should be noticed. We built a model (1:3) in the laboratory for a series of vibrating tests.
The experimental devices are the experimental cone crusher model with four URs driven by four identical motors (the type and specification is Y2-225M-6, 380 V, 50 Hz, 6-pole, -connection, rated power 7.5 kW, rated speed 980 r/min) shown in Fig. 17 (a), a series of testing instruments shown in Fig. 17 Fig. 18 and Fig. 19. As we can see from displacement data charts, the displacements in x-, y-and  -directions are in a state of unstable fluctuation at the beginning. After around 9 seconds, the experimental data of phase differences between IURs are given in Fig  and gradually approach 0, and the steady-state response of the  direction is at about 2.96°.
Through the comparison of the experiment and simulation analysis, we can verify the correctness of the theoretical research.

Conclusions
In this paper, through a series of theoretical research, we can stress some remarks as follows: (1) Using the improved small parameter average method, the dimensionless coupling equation of the crusher system is established.
(2) The parameter determination method of eccentric force offsetting self-synchronization under steady state of ultra-resonance is proposed by analyzing the steady-state phase differences, the synchronous capability curve and the stability curve. The parameter interval of the dimensionless parameters is designed: it is verified that the crusher has a set of steady-state phase differences solutions for idle state when e 1.414 2.5 r   ; the synchronous ability coefficient 12  increases in this interval and the  value is always greater than 10, indicating that the synchronization performance between motors in this parameter interval is well; the two sets of stability criteria are met simultaneously in the (1.414, 2.5) parameter interval.
(3) The design method of working crusher under steady-state of sub-resonance is proposed. It clarifies that the crusher adjusts the system frequency by increasing the materials and changes the phase differences between the four motors to adjust the movement of the body. The movement selection principle of the system is discussed. It is found material equivalent stiffness must not less than 1100 according to the system frequency ratio analysis. Three sets of coefficients are introduced to explore its variation with the material equivalent stiffness and the system damping ratio.
(4) The computer simulation and our experimental verifications fully prove the correctness of the theoretical research and the feasibility of crushing by periodic movement. Based on the self-synchronization of dual steady states theoretical model, a new type of crusher can be designed.

Acknowledgements
The authors would like to thank National Natural Science Foundation of China (Grant No. 51775094) for the financial support of the research projects.

Data Availability
The authors declare that datasets supporting the conclusions of this study are included within the paper.