# Synchronization of Two Eccentric Rotors Driven by One Motor with Two Flexible Couplings in a Spatial Vibration System

Authors: Yujia Li; Tao Ren; Jinnan Zhang; Minghong Zhang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

A dynamic model of a vibration system, in which eccentric rotors are driven by one motor with two flexible couplings, is developed in this study. The Lagrange equation is used to analyze the dynamic behavior of the vibration system. Synchronization theory and its motion law are investigated using Hamilton’s principle, and the validity of the theory is proven through numerical simulation and experimentation. Results show that the system has two synchronous motions, namely, 0 andπ phases. When the torsional stiffness difference between two flexible couplings on both sides of the motor or the resistance moment difference between two eccentric rotors increases, the eccentric rotors maintain the synchronization and stability of the vibration system by adjusting its phase difference. Synchronization theory and the analysis method of the flexible-drive vibration system are extended in this study. Moreover, the synchronous motion law of the vibration system based on bilateral flexible drive by one motor is revealed to provide guidance for the development of high-performance vibrating machines.


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## Body

## 1. Introduction

Synchronization is widespread; for example, satellites are required to operate synchronously with respect to the earth in their synchronous orbits, and people adjust the receiving frequency of their radios to the frequency emitted by the transmitting station to receive information from the radio station [1]. Owing to the development of science and technology, synchronization theory is now widely used in numerous industrial fields to improve social productive forces [2, 3].Synchronization theory of vibrating machines was established in the 1960s. Blehman installed two induction motors with eccentric mass and without forced connection on one vibrating body driving two inertial exciters. The inertial exciters can operate synchronously under certain conditions and remove the intermediate synchronous link of gear transmission to allow the machine to realize various vibration modes according to the working requirements [4].Two approaches, namely, increasing the number of excitation sources and increasing the exciting force, can be selected to improve the power and material processing efficiency of vibrating machines [5]. In the first approach, three motors are often used to drive the vibrating system. However, the vibrating system may present complex features, such as nonlinear mechanics and flexible body, when a vibration motor is added to the original vibrating machine driven by two motors. These features result in a double-equilibrium point of the system, which in turn causes power reduction of the vibrating machine. Therefore, selecting reasonable parameters (see glossary section) is necessary to avoid the double-equilibrium point [6, 7]. In addition, four motors are used for simultaneous simulation, which can achieve the equivalent vibration synchronization effect of two high-power vibration motors. However, adopting only the vibration synchronization method is difficult because the four eccentric rotors must have the same speed and phase to achieve the desired stable motion state of the vibration system. Existing research has proposed an approach that combines vibration and control synchronization, which increases the difficulty of design and application [8, 9].In the second approach, the exciting force of the vibration motor can be increased by increasing the mass of the eccentric rotors. Most shale shakers use short-shaped vibration motors to generate exciting force. The motor is installed in the middle of the supporting beam of the screen box. The supporting beam is welded on the lateral plate of the screen box, which bears the exciting force of the motor and transmits this force to the screen box. A long vibrating motor structure is formed by adding a lengthening cylinder on both sides of the short motor to improve the load conditions of the shale shaker with a short motor structure. The shale shaker lateral plates of the above two structures are subjected to alternating loads such as tension, compression, and bending. If the mass of the eccentric rotors continues to increase, then the screen box will fail due to the large alternating loads [10].Most domestic and foreign shale shaker products at present focus on the development of frequency conversion, variable trajectory, and intellectualization [11–13]. However, research on improving the exciting force and amplitude of a shale shaker to enhance its load conditions and increase its processing capacity is limited. Most existing synchronization theories are concerned with the following aspects: self-synchronization and control synchronization of rigid drive vibration systems with single and double vibro-bodies [14–16], self-synchronization of rigid drive vibration systems with parallel and cross axes [17–20], and self-synchronization of vibration systems flexibly driven by two motors in common axes [21]. However, related investigations on the spatial vibration and synchronization of vibration systems flexibly driven by one motor on both sides are few. Recent dynamic studies on eccentric rotors inspired the modeling in this study [22–24].In this work, a dynamic model of two eccentric rotors driven by one motor with two flexible couplings in a spatial vibration system is established. The synchronous motion law of the vibration system is investigated. Simulation and experiments verify the validity of the theory. The double-side, flexible-drive vibration motor structure and synchronization theory of the vibration system can be applied to engineering vibration machinery, such as oil-gas drilling shale shakers, shield vibrating screens, and vibration mills, to improve the mechanical stress and increase the material handling capacity.


## 2. Dynamic Behavior of an Eccentric Rotor Vibration System Driven by One Motor with Two Flexible Couplings

A structure of a vibration motor in which eccentric rotors are driven by one motor with two flexible couplings is proposed in this study (shown in Figure1) to maintain the good load conditions and synchronous-stable motion performance of vibrating machines under increasing exciting force. The short antivibration motor drives the two eccentric rotors on both sides by using two flexible couplings. The excitation parts of the long vibration motor are directly mounted on the lateral plates of the vibrating machine, and this considerably improves the load conditions of the machine [25, 26]. In the case of similar parameters, the ratio of the dynamic alternating bending moment of the vibration motor proposed in this work to that of the existing long-shaped vibration motor is 1:36 [27]. Accordingly, the reliability and strength of the proposed vibration motor and its vibrating machine are fully guaranteed.Figure 1
Schematic of the proposed long vibration motor.The proposed vibration motor is mounted on the vibrating body. The motor shaft drives the eccentric rotors to rotate periodically through the flexible couplings. The eccentric rotors drive the vibrating body to perform periodic forced vibration. On the one hand, the rotations of the eccentric rotors affect the motion law of the vibrating body. On the other hand, the translational and rotational motions of the vibrating body act as an implicated motion, which in turn acts on the eccentric rotors, thereby causing a change in the phase angle. Therefore, the eccentric rotors interact with the vibrating body. The vibration system includes a spatial vibration system of the vibrating body and a torsional vibration system of the eccentric rotors. Torsional vibration causes a phase difference between the eccentric rotors on both sides and thus affects the motion forms, such as plane, torsional, and composite vibration, of the vibrating body in space. Accordingly, the dynamic response of eccentric rotors has a crucial influence on the vibration of the vibrating body.As shown in Figure2, the dynamic model of a flexible-drive eccentric rotor system is obtained by simplifying the power source (short antivibration motor) and exciting components (eccentric rotors).Figure 2
Dynamic model of a flexible driven eccentric rotor system. 1 and 5: eccentric rotor; 2 and 4: flexible coupling; 3: motor rotor.The dynamic model is placed on the vibrating body, and a dynamic model of the vibration system is obtained, as shown in Figure3. The two sets of eccentric rotors are symmetrically placed with the same mass as m1 and the same eccentricity as e1. A coordinate system, XYZ, fixed to the inertial space is established with the center O of mass m in the initial statically balanced position. The X and Y distances from the rotational axis of the eccentric rotors to the mass center of the vibrating body are Lx and Ly, respectively. A motion coordinate system, xyz, is also established. The Z-direction distance from the origin of the coordinate system to the rotation center of two eccentric rotors is Lz, as shown in Figure 3(b).Figure 3
Dynamic model of two eccentric rotors driven by one motor with two flexible couplings in a spatial vibration system. (a) 3D schematic of the dynamic model. (b) Top view.
(a) (b)The existence of a phase difference between the two eccentric rotors causes the vibration of the body to rotate around theX, Y, and Z axes. The kinetic energy of the vibration system includes the kinetic energy of the plane motion in x and y directions, the kinetic energy of the rotational motion around the x, y, and z axes, and the kinetic energy of the motor and eccentric rotors. The potential energy of the vibration system includes the elastic potential energy of the flexible couplings, the gravitational potential energy of the eccentric rotors, and the elastic potential energy of the supporting springs of the vibrating body.The motion coordinates of the vibrating body (x, y, θx, θy, and θz), the rotational phase of the motor rotor (α0), and the rotational phase of the two eccentric rotors (α1 and α2) are selected as generalized coordinates. Given that the angular displacements (θx, θy, and θz) of the vibrating body around the x, y, and z axes are small, the high-order infinitesimal of the angular displacement and the term coupled with the rotation angles can be disregarded in the calculation.We assume thatsin⁡θx≈θx, sin⁡θy≈θy, sin⁡θz≈θz, cos⁡θx≈1, cos⁡θy≈1, and cos⁡θz≈1 [1]. The Lagrange equation is used to establish the differential motion equations of the vibration system in the x, y, θx, θy, and θz directions.(1)Mx¨+kxx+fxx˙=m1e1α˙12cos⁡α1+α¨1sin⁡α1+α˙22cos⁡α2+α¨2sin⁡α2My¨+kyy+fyy˙=m1e1α˙12sin⁡α1-α¨1cos⁡α1+α˙22sin⁡α2-α¨2cos⁡α2Jx+2m1Lz2+Ly2θ¨x+kθxθx+fθxθ˙x=m1e1Lz-α˙12sin⁡α1+α˙22sin⁡α2+α¨1cos⁡α1-α¨2cos⁡α2Jy+2m1Lz2+Lx2θ¨y+kθyθy+fθyθ˙y=m1e1Lzα˙12cos⁡α1-α˙22cos⁡α2+α¨1sin⁡α1-α¨2sin⁡α2Jz+2m1Lx2+Ly2θ¨z+kθzθz+fθzθ˙z=m1e1α˙12Lxsin⁡α1-Lycos⁡α1+α˙22Lxsin⁡α2-Lycos⁡α2+α¨1-Lxcos⁡α1-Lysin⁡α1+α¨2-Lxcos⁡α2-Lysin⁡α2The rotational power of the eccentric rotors is provided by the motor and transmitted through the flexible couplings. The moment balance equations of the main shaft of the vibration motor in the direction ofα0, α1, and α2 are obtained as follows:(2)JMα¨0+kα1α0-α1+kα2α0-α2+fα1α˙0-α˙1+fα2α˙0-α˙2=Mg-Mzm1e12+J0α¨1+kα1α1-α0+fα1α˙1-α˙0+m1ge1cos⁡α1=-m1e1y¨cos⁡α1-x¨sin⁡α1+m1e1θ˙z2Lycos⁡α1-Lxsin⁡α1+m1e1θzLy-Lxcos⁡α1-m1e1θzLx+Lysin⁡α1θ¨z-Mf1m1e12+J0α¨2+kα2α2-α0+fα2α˙2-α˙0+m1ge1cos⁡α2=-m1e1y¨cos⁡α2-x¨sin⁡α2+m1e1θ˙z2Lycos⁡α2-Lxsin⁡α2+m1e1θzLy-Lxcos⁡α2-m1e1θzLx+Lysin⁡α2θ¨z-Mf2The torsional vibration system of the eccentric rotors is analyzed, and the influence of the rotational motion and flexible connection of the antivibration motor on the response of the eccentric rotors is studied. The terms coupled with the body vibration are not considered temporarily.The fourth-order Runge–Kutta algorithm is used to numerically solve the second-order nonhomogeneous differential equations of the system. The specific parameters of the vibration system during simulation are as follows:m1 = 1.233 kg, e1 = 0.028 m, J0 = 0.01 kg∙m2, JM = 0.005 kg∙m2, kα1 = 20 N∙m/rad, kα2 = 18 N∙m/rad, fα1 = 0.1 N∙m/(rad/s), fα2 = 0.08 N∙m/(rad/s), Mf1 = Mf2 = 0.5, and Mz = 0.1. The initial values of each variable of the system are zero. Figure 4 is drawn from the numerical solution of (2).Figure 4
Synchronization simulation results of two eccentric rotors. (a) Angular velocity of two eccentric rotors. (b) Phase difference of two eccentric rotors.
(a) (b)A certain degree of small vibration is observed in each of the eccentric rotors on both sides when the system is started due to the slight difference in the stiffness of the couplings and the damping of the shafts on both sides. Subsequently, the rotors follow each other, and the system reaches a stable state in 1.5 s. After stabilization, the angular velocity curves of the two eccentric rotors coincide completely.As shown in Figure4, the motion of an eccentric rotor is a composite motion superimposed by a uniform rotational motion of the motor and its own small torsional vibration. Torsional vibration is a response caused by the periodic rotation of the motor, and its frequency is the rotation frequency of the motor. The eccentric rotors on both sides are driven by the same motor; therefore, their torsional vibration frequencies are the same and equal to the frequency of the motor rotor. The motor rotor rotates at 1400 r/min in steady state. Thus, the torsional vibration frequency is 23.3 Hz. Accordingly, when the two eccentric rotors are in steady motion, the phase difference is constant, and the rotors naturally reach the state of synchronization. The phase difference is −0.022 rad, which is close to zero phase synchronization, due to the slight difference in damping between the two eccentric rotor shafts. Compared with the conventional structure of two eccentric rotors driven by two coaxial motors [18], the proposed structure guarantees the synchronous motion of two eccentric rotors naturally; therefore, synchronization performance is unaffected by the motor speed difference. This structure can replace the conventional coaxial two-motor excitation structure and maintain the symmetry of the vibration system while increasing the power of the vibration motor, thus providing a new approach for coaxial vibration synchronization in engineering.Self-synchronization and stability conditions are required in the steady response for calculation; thus, the steady response of the vibration system is analyzed. The periodic rotational motion of the eccentric rotors is the main motion, and the small torsional vibration of the eccentric rotors is the auxiliary motion. Therefore, the angular velocity of the motor rotor can be set toω when the system is in steady operation. In practice, the angular velocity of rotors changes during a vibration period, but this change has little effect on the excitation force. This change can be ignored by referring to traditional synchronization theory, and the angular acceleration is set to zero [1]. Eccentric rotor 1 leads eccentric rotor 2 with phase difference Δα in constant speed rotation.The phase difference between the two eccentric rotors is an important factor that affects the self-synchronization and stability of a conventional rigid drive vibration system with two motors. A flexible connection and a phase difference exist between the motor shaft and the eccentric rotors in the proposed vibration system. However, the important factor that affects the self-synchronization and stability of the vibration system is the phase difference between the two eccentric rotors. Control parameterξis proposed in this study, makes the phase difference between the motor and eccentric rotors arbitrary and does not affect the stable phase difference △α between the two eccentric rotors. The relationship between angular displacements is as follows:(3)α0=ωtα1-α0=ξ△αα2-α0=ξ-1△αThe steady response of the vibration system in thex, y, θx, θy, and θz directions is(4)x=λxsin⁡ωt+γ1-φ1y=λysin⁡ωt+γ2-φ2θx=λθxsin⁡ωt+γ3-φ3θy=λθysin⁡ωt+γ4-φ4θz=λθzsin⁡ωt+γ5-φ5The amplitude is obtained by substituting (4) into (2).(5)λx=Akx-Mω22+fx2ω21/2=-Acos⁡φ1mxxω2λy=Bky-Mω22+fy2ω21/2=-Bcos⁡φ2myyω2λθx=Ckθx-JXω22+fθx2ω21/2=-Ccos⁡φ3Jxxω2λθy=Dkθy-JYω22+fθy2ω21/2=-Dcos⁡φ4Jyyω2λθz=Ekθz-JZω22+fθz2ω21/2=-Ecos⁡φ5Jzzω2The terms in (4) and (5) are listed in Appendix.


## 3. Synchronization Theory of the Eccentric Rotor Vibration System Driven by One Motor with Two Flexible Couplings

The two eccentric rotors of the proposed vibration system are not always in zero phase synchronization. The two eccentric rotors are occasionally in nonzero phase synchronization due to the asymmetry of the system structure, the unequal resistance moments of the two axes, or the unequal torsional stiffness of the couplings. Nonzero phase synchronization creates forces, resulting in a spatial composite motion in the vibration system. The steady response equation (4) must be substituted into the calculation for obtaining the self-synchronization and stability conditions to rationally design the parameters and structure of the vibration system.The vibration system is a complete nonconservative system under the action of exciting force, spring force, damping force, motor driving torque, and resistance torque. During a period of motion, the Hamilton action of the system is(6)H=∫0TLdt=1ω∫02πT-Vdωt=Mλx2ωπ2+Mλy2ωπ2+JXλθx2ωπ2+JYλθy2ωπ2+JZλθz2ωπ2+2πT0ω-kxλx2π2ω-kyλy2π2ω-kθxλθx2π2ω-kθyλθy2π2ω-kθzλθz2π2ω-kα1πωξ2·△α2-kα2πωξ-12△α2It is a complete nonconservative system. The driving torque of the motor isMg, the resistance torque of the motor is Mz, and the resistance torque of eccentric rotors 1 and 2 is Mf1 and Mf2, respectively. According to the Hamiltonian principle,(7)δH+1ω∫02π∑j=11Mjδqjdωt=0.With phase difference△α as the generalized coordinate and by substituting it into (3), the generalized force is obtained as(8)Mj=Mg-Mz∂α0∂△α+-Mf1∂α1∂△α+-Mf2∂α2∂△α=-ξMf1-ξ-1Mf2.By substituting (6) and (8) into (7), the synchronization equilibrium equation of the system can be obtained as follows:(9)S1sin⁡△α-2π△αωkα1ξ2+kα2ξ-12-2πξMf1+ξ-1Mf2ω=0,where S1 is the system synchronization index.(10)S1=e12m12πωkx-Mω2mxx2ω2+fx2+ky-Mω2myy2ω2+fy2+Jxω2Lz2-kθxLz2Jxx2ω2+fθx2+Jyω2Lz2-kθyLz2Jyy2ω2+fθy2+kθzLx2+Ly2-Jzω2Lx2+Ly2Jzz2ω2+fθz2.If the system parameters do not satisfy (9), no stable phase difference exists between the two eccentric rotors. Thus, the vibration system cannot operate synchronously.The conventional system (two eccentric rotors driven by two coaxial motors) requires the use of the same type of motor to achieve similar characteristics of the two motors and to ensure synchronous performance. If the two motors considerably differ in performance, the system may not be synchronized. The two eccentric rotors of the proposed vibration system are driven by the same motor; therefore, the driving torque of the motor is excluded in the synchronization equilibrium equation. In addition, the torsional stiffness of the flexible couplings is one of the factors that affect the self-synchronization conditions of the system.Assuming thatω = 157.08 rad/s, ξ = 0.5, and S1 = 10, the left side of (9) is assigned to g. When the ordinate of function g is 0, the value of abscissa Δα is the phase difference satisfying the self-synchronization condition. The torsional stiffness of the flexible couplings and the resistance moment of the eccentric rotors exert remarkable effects on the synchronous performance of the system. We assume that the parameters on both sides of the motor are the same. The self-synchronous condition diagram under four torsion stiffness values of the couplings (1, 50, 100, and 150 N∙m/rad) is shown in Figure 5. The ordinate g in Figure 5 is the left side of (9). The abscissa Δα in Figure 5 is the phase difference angle of two eccentric rotors. Three intersections exist between the curve and horizontal axis; therefore, the equation has two roots. One is called zero phase synchronization, which has an average value of 0 and is in the range of (-π/2,π/2), (3π/2,2π), or (-2π,-3π/2); the other is called π phase synchronization, which has an average value ofπand is in the range of (π/2,3π/2) or (-3π/2,-π/2). Accordingly, the system has two synchronization states of 0 and π phase.Figure 5
Self-synchronous condition diagram. (a)kα1 = kα2 = 1, 50, 100, and 150 N∙m/rad. (b) kα1 = kα2 = 600 N∙m/rad.
(a) (b)Figure5(a) shows that when the torsional stiffness of the couplings is small, the phase difference of the π phase synchronization is close to 180°. Further solving of the stable operation conditions is necessary to determine the synchronization state the system is in. The system can operate normally when the parameters meet self-synchronization and stability conditions. As shown in Figure 5(b), when the torsion stiffness of the couplings is 600 N∙m/rad, only one root is found in (9); thus, the system has only zero phase synchronization state. When the torsion stiffness of the couplings is infinite, the connection between the motor shaft and the eccentric rotor shaft becomes rigid and has a constant state of zero phase synchronization. Accordingly, the conventional rigid drive is a special case of the double-sided flexible drive proposed in this study.Assuming that the resistance moment of the eccentric rotors on both sides of the motor is the same, the self-synchronous condition diagram under four torsional stiffness difference (0, 50, 100, and 150 N∙m/rad) is shown in Figure 6(a). Results show that when the torsional stiffness difference between the two couplings is small, the phase difference of the π phase synchronization is close to 180°. When the torsional stiffness difference increases, the eccentric rotors automatically change the phase difference to maintain the self-synchronization and stability of the system.Figure 6
Self-synchronous condition diagram. (a)△kα = 0, 50, 100, and 150 N∙m/rad. (b) △Mf = 0, 50, 100, and 150 N∙m.
(a) (b)We assume that the torsional stiffness of the couplings on both sides of the motor is the same. The self-synchronous condition diagram under four resistance moment differences (0, 50, 100, and 150 N∙m) is shown in Figure 6(b). When the resistance moment difference between the two eccentric rotors is small, the phase difference of the 0 or π phase synchronization is close to 0° or 180°. When the resistance moment difference increases, the eccentric rotors automatically increase the phase difference to maintain the self-synchronization and stability of the system.In summary, the phase difference of the eccentric rotors is determined by the minimum value of the system’s average energy. When the torsional stiffness difference of the flexible couplings or the resistance moment difference of the eccentric rotors increases, the eccentric rotors adjust their phase difference to maintain the synchronization performance of the system.The stability condition of the system is obtained by the Hamilton principle.(11)∂2H∂△α2=S2cos⁡△α>0,where S2 is the system stability index.(12)S2=S1ω2πkα1ξ2+kα2ξ-12If the system parameters satisfyS2 > 0, then cos⁡△α > 0. The phase difference is stable within (-π/2,π/2), (3π/2,2π), or (-2π,-3π/2). If S2 < 0, then cos⁡△α < 0. The phase difference is stable within (π/2,3π/2) or (-3π/2,-π/2). Accordingly, the self-synchronization and stability conditions must be satisfied simultaneously to allow the system to work normally.


## 4. Synchronous Experimental Verification

A simulation analysis is conducted in this section to verify the validity of the dynamic model and synchronization theory investigated in this work. The simulation parameters are set approximate to the experimental parameters. Assuming that the parameters on both sides of the motor are the same, the simulation parameters are as follows:m1 = 1.233 kg, e1 = 0.028 m, M = 82.8 kg, J0 = 0.01 kg∙m2, Lx = 0.2105 m, Ly = 0.2645 m, Lz = 0.25 m, Jx = 10 kg∙m2, Jy = Jz = 20 kg∙m2, kx = ky = 100000 N/m, kθx = kθy = kθz = 90000 N∙m/rad, fx = fy = 600 N∙s/m, fθx = fθy = fθz = 400 N∙m/(rad/s), kα1 = kα2 = 10 N∙m/rad, fα1 = fα2 = 0.3 N∙m/(rad/s), and Mf1=Mf2. The simulation results are shown in Figure 7.Figure 7
Simulation results. (a) Displacement inx direction. (b) Displacement in y direction. (c) Angular displacement in z direction. (d) Steady trajectory of mass center.
(a) (b) (c) (d)The moment the motor starts, the displacement, and angular displacement of the vibrating body fluctuate in all directions then quickly enter the steady state within 1 s. The steady amplitudes of the vibrating body inx and y directions are equal. The motion trajectory of the mass center is a circle (Figure 7(d)). The phase difference between the two eccentric rotors is stable at 0 phase after a small fluctuation in the transitional stage. Accordingly, no rotation occurs around the x and y axes in the steady state of the vibration system. Given that the excitation axis deviates from the mass center, the vibrating body only rotates around the z axis (Figure 7(c)) and vibrates on the XOY plane (Figures 7(a) and 7(b)).A vibration testing platform is fabricated, as shown in Figure8, to verify the validity of the aforementioned analysis. The total vibration quality of the testing platform is 82.8 kg, and the exciting force is 2.23 kN. As shown in Figure 9, when the photoelectric switches detect the eccentric rotors, a low level is triggered; when no eccentric rotor is detected, a high level is triggered. The variation data of high and low levels are recorded by an oscilloscope to accurately measure the phase angles of the eccentric rotors. The synchronization process of the eccentric rotors is captured by a high-speed camera. The acceleration time-domain waveforms of the three-axis acceleration sensor and the measuring point trajectory are recorded by uTekSs data acquisition and analysis software. The synchronous motion law of the vibration system can be observed accurately by using the three testing and recording methods and verifying them against one another. This section verifies the two synchronous motion states of the system through two experiments.Figure 8
Experimental system.Figure 9
Prototype of the vibration motor.


### 4.1. Zero Phase Synchronization

The torsional stiffness of the flexible couplings is adjusted to be equal. Photoelectric switch 1 is connected to the CH3 channel, and photoelectric switch 2 is connected to the CH4 channel. When the motor is started, the two eccentric rotors follow each other until synchronization. As illustrated in Figure10, the data change frequency of CH3 and CH4 channels gradually increases in the initial stage of starting the motor. In steady state, high level “1” and low level “0” recorded by CH3 and CH4 channels, respectively, correspond exactly; hence, the two eccentric rotors are synchronized to zero phase. Figure 11 shows the zero phase synchronization process captured by the high-speed camera. The body is in translational vibration after synchronization. Figure 12 shows the acceleration of the test point in x, y, and z directions during the synchronization process, and Figure 13 presents the steady trajectory of the test point.Figure 10
Zero phase synchronization process recorded by the oscilloscope.Figure 11
Zero phase synchronization process captured by the high-speed camera.Figure 12
Acceleration of the test point inx, y, and z directions.Figure 13
Steady trajectory of the test point.As shown in Figure12, the acceleration of the vibrating body fluctuates when the motor starts then quickly enters the steady state. A slight periodic fluctuation is observed in the steady acceleration of the vibrating body in the z direction. The simulation results in Figure 7(d) show that the steady trajectory on the xoy plane is a circle due to the similar parameters on both sides of the motor. However, the steady trajectory on the xoy plane is a slightly tilted ellipse, and the steady trajectory on the yoz plane is a slightly tilted line in the experimental results in Figure 13. This phenomenon is due to the unequal resistance moments in the experiment, which result in a slight phase difference between the two eccentric rotors. The phase difference induces the torsional vibration of the system.


### 4.2.π Phase Synchronization

The torsional stiffness of the flexible couplings is adjusted to be unequal. When the motor is started, the two eccentric rotors follow each other until synchronization. As illustrated in Figure14, the data change frequency of CH3 and CH4 channels gradually increases in the initial stage of starting the motor. In steady state, the high level “1” recorded by the CH3 channel corresponds to the low level “0” recorded by the CH4 channel; hence, the two eccentric rotors are synchronized to π phase. Figure 15 shows the π phase synchronization process captured by the high-speed camera. The body is in torsional vibration after synchronization. Figure 16 presents the steady trajectory of the test point.Figure 14
π phase synchronization process recorded by the oscilloscope.Figure 15
π phase synchronization process captured by the high-speed camera.Figure 16
Steady trajectory of the test point.Figure16 shows that the synchronization state of the eccentric rotors is close to that of π phase synchronization. The steady trajectory on the yoz plane is a slightly tilted ellipse, and the inclination angle is 6°. This condition can be attributed to the unequal resistance moments in the experiment, which result in a slight phase difference between the two eccentric rotors. The eccentric rotors automatically adjust the phase difference to maintain the self-synchronization and stability performance of the system. The two experiments are verified in the analysis in Section 3. Two synchronous motion states of the vibration system, namely, 0 and π phase, are investigated in this study.In the experiment, the vibration direction angle of the vibration system on the YZ plane is within the range of 6° from the theoretical simulation result. This finding can be attributed to the unequal eccentric rotor shaft damping on both sides of the motor in the experiment, thus creating a slight phase difference. The resistance torque and phase difference cannot be accurately measured due to the limitations in experimental conditions. In the experiment, an initial attempt is made to apply a small resistance torque on the axis of the eccentric rotor with minimal damping. The phase difference of the two eccentric rotors and the long axis inclination of the elliptical motion track become zero. Overall, a resistance torque can be applied to the axis of the eccentric rotor with minimal damping, resulting in equal resistance torques of the two shafts. In addition, the machining and assembly accuracy of the shaft must be ensured to reduce the phase difference.


## 4.1. Zero Phase Synchronization

The torsional stiffness of the flexible couplings is adjusted to be equal. Photoelectric switch 1 is connected to the CH3 channel, and photoelectric switch 2 is connected to the CH4 channel. When the motor is started, the two eccentric rotors follow each other until synchronization. As illustrated in Figure10, the data change frequency of CH3 and CH4 channels gradually increases in the initial stage of starting the motor. In steady state, high level “1” and low level “0” recorded by CH3 and CH4 channels, respectively, correspond exactly; hence, the two eccentric rotors are synchronized to zero phase. Figure 11 shows the zero phase synchronization process captured by the high-speed camera. The body is in translational vibration after synchronization. Figure 12 shows the acceleration of the test point in x, y, and z directions during the synchronization process, and Figure 13 presents the steady trajectory of the test point.Figure 10
Zero phase synchronization process recorded by the oscilloscope.Figure 11
Zero phase synchronization process captured by the high-speed camera.Figure 12
Acceleration of the test point inx, y, and z directions.Figure 13
Steady trajectory of the test point.As shown in Figure12, the acceleration of the vibrating body fluctuates when the motor starts then quickly enters the steady state. A slight periodic fluctuation is observed in the steady acceleration of the vibrating body in the z direction. The simulation results in Figure 7(d) show that the steady trajectory on the xoy plane is a circle due to the similar parameters on both sides of the motor. However, the steady trajectory on the xoy plane is a slightly tilted ellipse, and the steady trajectory on the yoz plane is a slightly tilted line in the experimental results in Figure 13. This phenomenon is due to the unequal resistance moments in the experiment, which result in a slight phase difference between the two eccentric rotors. The phase difference induces the torsional vibration of the system.


## 4.2.π Phase Synchronization

The torsional stiffness of the flexible couplings is adjusted to be unequal. When the motor is started, the two eccentric rotors follow each other until synchronization. As illustrated in Figure14, the data change frequency of CH3 and CH4 channels gradually increases in the initial stage of starting the motor. In steady state, the high level “1” recorded by the CH3 channel corresponds to the low level “0” recorded by the CH4 channel; hence, the two eccentric rotors are synchronized to π phase. Figure 15 shows the π phase synchronization process captured by the high-speed camera. The body is in torsional vibration after synchronization. Figure 16 presents the steady trajectory of the test point.Figure 14
π phase synchronization process recorded by the oscilloscope.Figure 15
π phase synchronization process captured by the high-speed camera.Figure 16
Steady trajectory of the test point.Figure16 shows that the synchronization state of the eccentric rotors is close to that of π phase synchronization. The steady trajectory on the yoz plane is a slightly tilted ellipse, and the inclination angle is 6°. This condition can be attributed to the unequal resistance moments in the experiment, which result in a slight phase difference between the two eccentric rotors. The eccentric rotors automatically adjust the phase difference to maintain the self-synchronization and stability performance of the system. The two experiments are verified in the analysis in Section 3. Two synchronous motion states of the vibration system, namely, 0 and π phase, are investigated in this study.In the experiment, the vibration direction angle of the vibration system on the YZ plane is within the range of 6° from the theoretical simulation result. This finding can be attributed to the unequal eccentric rotor shaft damping on both sides of the motor in the experiment, thus creating a slight phase difference. The resistance torque and phase difference cannot be accurately measured due to the limitations in experimental conditions. In the experiment, an initial attempt is made to apply a small resistance torque on the axis of the eccentric rotor with minimal damping. The phase difference of the two eccentric rotors and the long axis inclination of the elliptical motion track become zero. Overall, a resistance torque can be applied to the axis of the eccentric rotor with minimal damping, resulting in equal resistance torques of the two shafts. In addition, the machining and assembly accuracy of the shaft must be ensured to reduce the phase difference.


## 5. Conclusions

The dynamic behavior and synchronization of the two eccentric rotors driven by one motor with two flexible couplings in a spatial vibration system are investigated in this study. First, the torsional stiffness and damping of the flexible couplings are considered. A dynamic model of the two eccentric rotors driven by one motor with two flexible couplings in a spatial vibration system is established by using the Lagrange method. Second, a control parameter is introduced, and the steady responses of the system are solved. Third, the synchronization and stability conditions of the system are derived using the Hamilton principle. Fourth, the fourth-order Runge–Kutta algorithm is used to numerically solve the differential equations of the vibration system. The displacement and motion trajectory of the vibrating body are also simulated and analyzed. Finally, a vibration testing platform is fabricated, and the validity of the theoretical calculation is verified through experiments.The analysis shows that the eccentric rotor rotates periodically with the motor and superimposes a small amount of torsional vibration. The torsional vibration frequency is equal to the motor rotation frequency. The steady phase difference between the two eccentric rotors is constant; thus, the synchronous motion state is naturally reached. By analyzing the synchronization conditions, the system is found to have two synchronous motion states, namely, 0 andπ phases. In the synchronous steady state, the phase difference between the two eccentric rotors is determined by the minimum value of the average energy of the system. When the torsional stiffness difference of the flexible couplings or the resistance torque difference of the eccentric rotors increases, the eccentric rotors adjust their phase difference to maintain the synchronization performance of the system. The double-sided flexible-drive vibration motor structure and the synchronization theory of its vibration system investigated in this study can be applied not only to oil-gas drilling engineering but also to shield, mining, and environmental protection engineering. Owing to the limitation in experimental conditions, this work preliminary tests several parameters that affect the synchronization performance of the vibration system. In the future, other parameters, such as resistance torque, torsional stiffness, and phase difference, will be accurately measured.



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*Source: 2969687-2019-04-07.xml*