# Coupling Synchronization Criterion of Two Hydraulic Motors in an Eccentric Rotary Vibration Machine

Authors: Chun-lei Luo; Xin Mo; Jin-yang Li; Zhi-qing Tang; Song-song Huang
Journal: Shock and Vibration
 (2019)

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## Abstract

In an eccentric rotating system driven by two hydraulic motors without synchronous gears, vibration coupling may help render motion stable. In order to investigate how vibration coupling influences the motion, the coupling characteristics of the vibration system were studied regarding the differences between two motors such as leakage network, coulomb damping network, and pressure loss network, and the sensitivity of the influence factors was also studied. The influence of tiny differences between the two motors, tiny differences in the motion pair structure, in the oil temperature and in the resistance coefficient on the coupling motion were discovered, and the criterion for synchronous motion were obtained consequently. The results show that the influence of the resistance coefficient difference on system motion stability is the greatest, accounting for 46.7%, and the influence of the difference in motion pair structure (e.g. motor piston clearance) is the second, accounting for 32.8%. For motors with displacement 80 ml/r, the condition of self-synchronization is that the difference in piston clearance between the two motors is equal to or smaller than 6μm. Experiments have proved the correctness of the theory and showed that the synchronization can be achieved by leakage compensation, damping compensation, and back-pressure compensation of the external system by means of control when the motors rotate slowly enough for system response. The study shows that the coupling synchronous model can reduce the force of the gear for the eccentricity rotary system with synchronous gear, and that the synchronous stability can be improved for the eccentricity rotary system without synchronous gear.


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## Body

## 1. Introduction

The theoretical system and technical application of the vibration synchronization and control synchronization of the mechanical system driven by electromotors have long been developed, and even the nonlinear problems of the mechanical system such as vibration stability and bifurcation of self-synchronous systems driven by electromotors have been well solved [1, 2]. However, because of the limited driving capacity of the electromotor-driven system, it is difficult to obtain the application of synchronous vibration and control motor transmission under large loads.Hydraulic-driven eccentric rotary vibration system is widely used. It is characterized by high power and a large resistance coefficient and stiffness of working object. And, its synchronization is constrained by the synchronizing gear. The problems of the system are as follows: gear driving mechanical structure is too complicated; gears and other transmission parts are prone to damage etc [3]. Field tests show that the coupling effect has a direct influence on the gear stress and vibrating noise of the two rotating systems. The study of synchronization can provide theoretical basis for reducing influences of nonsynchronized factors on synchronizing gear stress, and even for removing the synchronization gears and simplifying the structure.Reference [4] studied the effects of friction torque, dynamic stiffness, dynamic damping, structural parameters of vibrator, hydraulic motor leakage, and other factors on the stable working process and transition process of the system. References [1, 2, 5, 6] researched and applied coupling synchronization mechanism driven by motors. Reference [7] reported a preliminary study about the basic problems on hydraulic motor-driven synchronization system with small and unchangeable damping coefficient and stiffness, and in the modeling process, electrohydraulic control was neglected. Considering high power and a large damping coefficient, references [3, 4] studied electrohydraulic coupling modeling and simulation about the eccentricity gyration hydraulic-driven control system, and it discovered that the two motors in the eccentric gyration system possess coupling synchronization performance when the hydraulic system is under certain conditions (such as two systems with the same mass flow of hydraulic oil, hydraulic motors with required divulging differences, gyration systems with required mechanical efficiency differences, foundation soil with elastic stiffness in required range, etc). In reference [8], considering the soil hysteresis characteristics, the influence of working parameters of hydraulic vibration hammer on coupling characteristics was studied. Also, the synchronicity, stability, and sensitivity of the hydraulic vibration hammer without synchronous gear were studied in reference [8], considering the leakage characteristics of hydraulic system.In this paper, based on the analysis of mechanical-electric-hydraulic coupling characteristics of the two rotating systems in dynamic balance process, the synchronous criterion of steady vibration eccentric gyration hydraulic-driven system was studied considering the factors such as mechanical damping, load damping, hydraulic system viscosity, hydraulic motor motion pair clearance, leakage, etc. In order to avoid the adverse effect of pipeline differences on synchronization, the electrohydraulic proportional valve was directly fixed at the inlet of the hydraulic motor. The coupled synchronous vibration hammer model is shown in Figure1 [9, 10].Figure 1
New vibration piling hammer-soil model. (1), (2) Eccentric blocks 1 and 2. (3) Pile hammer body. (4) Foundation soil’s flexibility. (5) Foundation soil’s damping. (6), (7) Hydraulic motors 6 and 7. (8), (9) Electrohydraulic proportional valves 8 and 9.


## 2. Description of the Two Rotating Systems in Dynamic Equilibrium Process

How to achieve synchronous motion of hydraulic motors has been restricting the application of self-synchronization of hydraulic motor in vibration engineering. But, there are more factors affecting the self-synchronization of hydraulic motors than electric motors [11–13], which is determined by the characteristics of the hydraulic system itself, such as flow leakage, large damping, and so on. Our research mainly includes two parts: coupling and compensation. Synchronization criterion is found by kinematic coupling, and the deficiencies of vibration system are compensated by means of control, such as reducing the difference between two hydraulic motors and other parts in the system (manufacturing difference or wear difference after using for a period of time) to meet the synchronization criterion and achieve synchronization. This article mainly focuses on the former (synchronization criterion and its influencing factors).The hydraulic flow entering the working chambers of the two motors is approximately the same under the action of the flow control valve. As shown in Figure2, in the working process of two hydraulic motors, when there is a phase difference between the two eccentric rotary systems, the hydraulic motor with advanced phase is hindered by the vibration resistance moment of the hydraulic motor with hysteretic phase, which leads to the increase of the load and causes a deceleration movement occurs on the motor with the advanced phase. Meanwhile, the increase of the load pressure causes the increase of the leakage of the motor; so, the effective working flow of the motor is relatively decreased, which further reduces the speed of motor with the advanced phase. At the same time, the opposite occurs to the hydraulic motor with the hysteretic phase. Simply put, the eccentric rotor with the advanced phase helps accelerate the eccentric rotor with the hysteretic phase, until both reach the same speed. Next, this paper will explore under what conditions the two hydraulic motors can achieve synchronous motion.Figure 2
Dynamic balancing process of the two rotary systems in the hydraulic system.


## 3. The Derivation of Coupling Characteristic Solution

### 3.1. Vibration Coupling Model

The structure principle of inertia-excitation gyration with two motors is shown in Figure1. The hydraulic power component is quantitative plunger pump. The hydraulic control component is Model BT32, which is a high-pressure and great flow capacity proportion flow valve and is forerunner control type with displacement-electricity feedback. The hydraulic actuator is quantitative plunger motor [3, 14]. It is assumed that the quality of eccentric block, eccentric radius, and moment of inertia in the gyration system is the same. The differences of hydraulic motor leak of each gyration system should not be ignored. Vibration power consumption, damping force, and stiffness of the object should not be ignored [15–17]. The precision of electrohydraulic proportional flow valve is sufficient [3].Luo [3] studied the coupling relationship about vibrating of the hydraulic-driven system:(1)12mr2Δα¨1+f1−mra3ω01Δα˙1+mra2ω02sinγ−a4ω02cosγΔα˙2+f1ω01+mr2a2ω012sinγ−a3ω012−a4ω022cosγ−μ1P1q=0,12mr2Δα¨2−mra1ω01sinγ+a3ω01cosγΔα˙1+f2−mra4ω02Δα˙2+f2ω02−mr2a1ω012sinγ+a4ω022+a3ω012cosγ−μ2P2q=0.In equation (1),(2)a1=mrky−Mα˙12α˙12+fyα˙1+Mα¨1α¨1ky−Mα˙122+fyα˙1+Mα¨12,a2=mrky−Mα˙22α˙22+fyα˙2+Mα¨2α¨2ky−Mα˙222+fyα˙2+Mα¨22,a3=mrky−Mα˙12α˙12−fyα˙1+Mα¨1α¨1ky−Mα˙122+fyα˙1+Mα¨12,a4=mrky−Mα˙22α˙22−fyα˙2+Mα¨2α¨2ky−Mα˙222+fyα˙2+Mα¨22.The equations (1) and (2) has the coupling items: α1, α2, Δα1, and Δα2, which are symmetrically distributed in the dynamic formula, indicating that the minor variation of rotational motion in the two rotating systems are mutually limited. The value of ai (i = 1–4) is determined by the structure parameters of the vibration system and external conditions, such as M (mass of the whole vibration system), Ky (synthesis stiffness of foundation soil and isolation spring) [18], and so on, which indicate the influence of the oscillatory system on the strength of the vibration coupling between the exciters.


### 3.2. Mathematical Model of Leakage of Hydraulic Motor

As shown in Figure3, the internal leakage of the motor is determined mainly by three factors in the working operation of the slanting axial piston motor, which are the plunger pair leakage, the slipper pair leakage, and the Port Plate leakage, respectively [19–21].Figure 3
Internal leakage diagram of the axial piston motor.The piston pair has a matching gap and there is a high-pressure difference between the two ends of the piston ring. When the plunger moves to and from, the oil liquid in the column plug will leak from the piston pair clearance to the motor shell under the action of pressure difference. Figure4 shows a schematic of the plunger pair.Figure 4
Plunger pair.The instantaneous leakage flow of a single piston pair is(3)Ql1i=πdδ1312ηl1+ε2PLi−πdδ12vi=1,2,v=−D2sinθψ˙isinψii=1,2.The leakage of the motor is mainly caused by the leakage of the piston pair, but the leakage of the Port Plate and the slipper should not be ignored either. The leakage flow of the slipper pair is as follows:(4)Ql2i=πδ236ηInR2/R1⋅λPLii=1,2.The leakage flow of the Port Plate as follows:(5)Ql3i=πδ336η1InR6/R8+1InR7/R5φ2−φ1⋅λPLii=1,2.The total leakage of oil in slanting axial piston motor is as follows:(6)QLi=Ql1i+Ql2i+Ql3i.


### 3.3. The Derivation of Coupling Characteristic Solution

Hydraulic pressure energy drives eccentric block to rotate, and then convert into mechanical energy. The energy conversion relationship is as follows:(7)piQi=Tiα˙i.α˙i=ω0i+Δα˙ii=1,2.In the equation above,pi stands for the oil pressure, Qi for effective instantaneous flow, and α˙i for the instantaneous angular velocity.(8)Ti=piQiα˙i,i=1,2.Because the electrohydraulic proportional valve was directly fixed at the inlet of the hydraulic motor, under the control of the proportional valve, the working flow into the two motors is equal; so, high accuracy of angular velocityω01, ω02 can be achieved (the range of error can be controlled within 2%). Therefore, it can be assumed that(9)ω01=ω02=ω0.Substituting equations (8) and (9) in equation (1) gives(10)Ef1=2f1mr2,Ef2=2f2mr2,Eμ1=2μ1mω0r2Eμ2=2μ2mω0r2,Ek=2a2ω0r,Ec=2a3ω0r.Then, we can get(11)Δα¨1+Ef1−Ec+Eμ1ω0Q1p1Δα˙1+Eksinγ−EccosγΔα˙2+12Ekω0sinγ+Ef1ω0−12Ecω0−12Ecω0cosγ−Eμ1Q1p1=0,Δα¨2−Eksinγ+EccosγΔα˙1+Ef2−Ec+Eμ2ω0Q2p2Δα˙2−12Ekω0sinγ+Ef2ω0−12Ecω0−12Ecω0cosγ−Eμ2Q2p2=0.In equation (10), under the condition of certain structural parameters about vibration pile and certain angular frequency ω0, Ef1 and Ef2 are only determined by mechanical transfer resistance coefficient of gyration systems 1 and 2; Eμ1 and Eμ2 are only determined by mechanical efficiency of the hydraulic motor; Ek is only related to stiffness of foundation soil; and Ec is only related to damping of foundation soil.The equation (11) describes the changes about microangle Δα1, Δα2 within a cycle. Suppose the foundation soil characteristic parameters are non-time-variable. Under the condition that gyration speed ω0 is approximately constant, it is a second-order nonlinear equation with constant coefficients. Considering the periodic motion within one cycle, the singularity equation in the phase plane (Δα1, Δα2) consisted by equation disaggregation is(12)Δα¨1+Ef1−Ec+Eμ1ω0Q1p1Δα˙1+Eksinγ−EccosγΔα˙2+12Ekω0sinγ+Ef1ω0−12Ecω0−12Ecω0cosγ−Eμ1Q1p1=0,Δα¨2−Eksinγ+EccosγΔα˙1+Ef2−Ec+Eμ2ω0Q2p2Δα˙2−12Ekω0sinγ+Ef2ω0−12Ecω0−12Ecω0cosγ−Eμ2Q2p2=0.Disappearing items of equations (12) and symmetrical reorganizing, we can get the solution to the equation as follows:(13)Δα˙1=Δα˙′¯H+Δα˙′H,Δα˙2=Δα˙′¯H−Δα˙′H.In equation (13),(14)H=Ek2sin2γ−Ec2cos2γ+Ef1Ef2−2EcEf¯+Ec2+1ω02Ef¯N¯−ΔEf2ΔN−2EN¯c+N1N2ω02,Δα˙′¯=−12Ek2sin2γω0+12Ec2ω0cos2γ−14Ekω0ΔEfsinγ−Ef1Ef2ω0−12Ec2ω012Ecω0cosγEf¯+32Ecω0Ef¯+34EksinγΔN+32EcN¯cosγ−12EcN¯+N1N2ω0,Δα˙′=12Ekω0Ef¯sinγ+14Ecω0ΔEfcosγ+14Ecω0ΔEf−32EkN¯sinγ−34EcΔNcosγ−ΔEfN¯+Ef¯ΔN−34EcΔN,In equation (14),(15)ΔEf=Ef1−Ef2,Ef¯=Ef1+Ef22Ef1=Ef¯+ΔEf2,Ef2=Ef¯−ΔEf2,(16)N¯=N1+N22,ΔN=N1−N2,and(17)N1=Eμ1Q1p1=N¯+ΔN2,N2=Eμ2Q2p2=N¯−ΔN2.Equations (13) and (14) include not only the situation of the power consumed by the eccentric block at any position within one circle, but also the symmetry of the angular velocity of eccentric gyration between systems 1 and 2, and the mutual energy consumption characteristics.In equation (17), Q1, Q2 are the effective instantaneous flow of the hydraulic motors 1 and 2, which constitute the total flow of the hydraulic motor with the leakage flow through the electromagnetic proportional valve to the hydraulic motor.(18)Q1=Q−QL1,Q2=Q−QL2,then,(19)N1=Eμ1P1Q−Eμ1P12wδ113+uδ123+vδ133,N2=Eμ2P2Q−Eμ2P22wδ213+uδ223+vδ233,ΔN=Eμ1P1−Eμ2P2Q−Eμ1P12wδ113+uδ123+vδ133+Eμ2P22wδ213+uδ223+vδ233,N¯=12Eμ1P1+Eμ2P2Q−Eμ1P12wδ113+uδ123+vδ133−Eμ2P22wδ213+uδ223+vδ233,wherein(20)w=πd12ηl1+ε2,u=π6ηInR2/R1⋅λ,v=π6η1InR6/R8+1InR7/R5φ2−φ1⋅λ.


## 3.1. Vibration Coupling Model

The structure principle of inertia-excitation gyration with two motors is shown in Figure1. The hydraulic power component is quantitative plunger pump. The hydraulic control component is Model BT32, which is a high-pressure and great flow capacity proportion flow valve and is forerunner control type with displacement-electricity feedback. The hydraulic actuator is quantitative plunger motor [3, 14]. It is assumed that the quality of eccentric block, eccentric radius, and moment of inertia in the gyration system is the same. The differences of hydraulic motor leak of each gyration system should not be ignored. Vibration power consumption, damping force, and stiffness of the object should not be ignored [15–17]. The precision of electrohydraulic proportional flow valve is sufficient [3].Luo [3] studied the coupling relationship about vibrating of the hydraulic-driven system:(1)12mr2Δα¨1+f1−mra3ω01Δα˙1+mra2ω02sinγ−a4ω02cosγΔα˙2+f1ω01+mr2a2ω012sinγ−a3ω012−a4ω022cosγ−μ1P1q=0,12mr2Δα¨2−mra1ω01sinγ+a3ω01cosγΔα˙1+f2−mra4ω02Δα˙2+f2ω02−mr2a1ω012sinγ+a4ω022+a3ω012cosγ−μ2P2q=0.In equation (1),(2)a1=mrky−Mα˙12α˙12+fyα˙1+Mα¨1α¨1ky−Mα˙122+fyα˙1+Mα¨12,a2=mrky−Mα˙22α˙22+fyα˙2+Mα¨2α¨2ky−Mα˙222+fyα˙2+Mα¨22,a3=mrky−Mα˙12α˙12−fyα˙1+Mα¨1α¨1ky−Mα˙122+fyα˙1+Mα¨12,a4=mrky−Mα˙22α˙22−fyα˙2+Mα¨2α¨2ky−Mα˙222+fyα˙2+Mα¨22.The equations (1) and (2) has the coupling items: α1, α2, Δα1, and Δα2, which are symmetrically distributed in the dynamic formula, indicating that the minor variation of rotational motion in the two rotating systems are mutually limited. The value of ai (i = 1–4) is determined by the structure parameters of the vibration system and external conditions, such as M (mass of the whole vibration system), Ky (synthesis stiffness of foundation soil and isolation spring) [18], and so on, which indicate the influence of the oscillatory system on the strength of the vibration coupling between the exciters.


## 3.2. Mathematical Model of Leakage of Hydraulic Motor

As shown in Figure3, the internal leakage of the motor is determined mainly by three factors in the working operation of the slanting axial piston motor, which are the plunger pair leakage, the slipper pair leakage, and the Port Plate leakage, respectively [19–21].Figure 3
Internal leakage diagram of the axial piston motor.The piston pair has a matching gap and there is a high-pressure difference between the two ends of the piston ring. When the plunger moves to and from, the oil liquid in the column plug will leak from the piston pair clearance to the motor shell under the action of pressure difference. Figure4 shows a schematic of the plunger pair.Figure 4
Plunger pair.The instantaneous leakage flow of a single piston pair is(3)Ql1i=πdδ1312ηl1+ε2PLi−πdδ12vi=1,2,v=−D2sinθψ˙isinψii=1,2.The leakage of the motor is mainly caused by the leakage of the piston pair, but the leakage of the Port Plate and the slipper should not be ignored either. The leakage flow of the slipper pair is as follows:(4)Ql2i=πδ236ηInR2/R1⋅λPLii=1,2.The leakage flow of the Port Plate as follows:(5)Ql3i=πδ336η1InR6/R8+1InR7/R5φ2−φ1⋅λPLii=1,2.The total leakage of oil in slanting axial piston motor is as follows:(6)QLi=Ql1i+Ql2i+Ql3i.


## 3.3. The Derivation of Coupling Characteristic Solution

Hydraulic pressure energy drives eccentric block to rotate, and then convert into mechanical energy. The energy conversion relationship is as follows:(7)piQi=Tiα˙i.α˙i=ω0i+Δα˙ii=1,2.In the equation above,pi stands for the oil pressure, Qi for effective instantaneous flow, and α˙i for the instantaneous angular velocity.(8)Ti=piQiα˙i,i=1,2.Because the electrohydraulic proportional valve was directly fixed at the inlet of the hydraulic motor, under the control of the proportional valve, the working flow into the two motors is equal; so, high accuracy of angular velocityω01, ω02 can be achieved (the range of error can be controlled within 2%). Therefore, it can be assumed that(9)ω01=ω02=ω0.Substituting equations (8) and (9) in equation (1) gives(10)Ef1=2f1mr2,Ef2=2f2mr2,Eμ1=2μ1mω0r2Eμ2=2μ2mω0r2,Ek=2a2ω0r,Ec=2a3ω0r.Then, we can get(11)Δα¨1+Ef1−Ec+Eμ1ω0Q1p1Δα˙1+Eksinγ−EccosγΔα˙2+12Ekω0sinγ+Ef1ω0−12Ecω0−12Ecω0cosγ−Eμ1Q1p1=0,Δα¨2−Eksinγ+EccosγΔα˙1+Ef2−Ec+Eμ2ω0Q2p2Δα˙2−12Ekω0sinγ+Ef2ω0−12Ecω0−12Ecω0cosγ−Eμ2Q2p2=0.In equation (10), under the condition of certain structural parameters about vibration pile and certain angular frequency ω0, Ef1 and Ef2 are only determined by mechanical transfer resistance coefficient of gyration systems 1 and 2; Eμ1 and Eμ2 are only determined by mechanical efficiency of the hydraulic motor; Ek is only related to stiffness of foundation soil; and Ec is only related to damping of foundation soil.The equation (11) describes the changes about microangle Δα1, Δα2 within a cycle. Suppose the foundation soil characteristic parameters are non-time-variable. Under the condition that gyration speed ω0 is approximately constant, it is a second-order nonlinear equation with constant coefficients. Considering the periodic motion within one cycle, the singularity equation in the phase plane (Δα1, Δα2) consisted by equation disaggregation is(12)Δα¨1+Ef1−Ec+Eμ1ω0Q1p1Δα˙1+Eksinγ−EccosγΔα˙2+12Ekω0sinγ+Ef1ω0−12Ecω0−12Ecω0cosγ−Eμ1Q1p1=0,Δα¨2−Eksinγ+EccosγΔα˙1+Ef2−Ec+Eμ2ω0Q2p2Δα˙2−12Ekω0sinγ+Ef2ω0−12Ecω0−12Ecω0cosγ−Eμ2Q2p2=0.Disappearing items of equations (12) and symmetrical reorganizing, we can get the solution to the equation as follows:(13)Δα˙1=Δα˙′¯H+Δα˙′H,Δα˙2=Δα˙′¯H−Δα˙′H.In equation (13),(14)H=Ek2sin2γ−Ec2cos2γ+Ef1Ef2−2EcEf¯+Ec2+1ω02Ef¯N¯−ΔEf2ΔN−2EN¯c+N1N2ω02,Δα˙′¯=−12Ek2sin2γω0+12Ec2ω0cos2γ−14Ekω0ΔEfsinγ−Ef1Ef2ω0−12Ec2ω012Ecω0cosγEf¯+32Ecω0Ef¯+34EksinγΔN+32EcN¯cosγ−12EcN¯+N1N2ω0,Δα˙′=12Ekω0Ef¯sinγ+14Ecω0ΔEfcosγ+14Ecω0ΔEf−32EkN¯sinγ−34EcΔNcosγ−ΔEfN¯+Ef¯ΔN−34EcΔN,In equation (14),(15)ΔEf=Ef1−Ef2,Ef¯=Ef1+Ef22Ef1=Ef¯+ΔEf2,Ef2=Ef¯−ΔEf2,(16)N¯=N1+N22,ΔN=N1−N2,and(17)N1=Eμ1Q1p1=N¯+ΔN2,N2=Eμ2Q2p2=N¯−ΔN2.Equations (13) and (14) include not only the situation of the power consumed by the eccentric block at any position within one circle, but also the symmetry of the angular velocity of eccentric gyration between systems 1 and 2, and the mutual energy consumption characteristics.In equation (17), Q1, Q2 are the effective instantaneous flow of the hydraulic motors 1 and 2, which constitute the total flow of the hydraulic motor with the leakage flow through the electromagnetic proportional valve to the hydraulic motor.(18)Q1=Q−QL1,Q2=Q−QL2,then,(19)N1=Eμ1P1Q−Eμ1P12wδ113+uδ123+vδ133,N2=Eμ2P2Q−Eμ2P22wδ213+uδ223+vδ233,ΔN=Eμ1P1−Eμ2P2Q−Eμ1P12wδ113+uδ123+vδ133+Eμ2P22wδ213+uδ223+vδ233,N¯=12Eμ1P1+Eμ2P2Q−Eμ1P12wδ113+uδ123+vδ133−Eμ2P22wδ213+uδ223+vδ233,wherein(20)w=πd12ηl1+ε2,u=π6ηInR2/R1⋅λ,v=π6η1InR6/R8+1InR7/R5φ2−φ1⋅λ.


## 4. Synchronous Criterion

The necessary criterion for synchronization is that there must exist a solutionγ to equation (13), and it makes(21)Δα˙′=0.That is to say, the solution to equation (22) must exist [22].(22)12Ekω0Ef¯sinγ+14Ecω0ΔEfcosγ+14Ecω0ΔEf−32EkN¯sinγ−34EcΔNcosγ−ΔEfN¯+Ef¯ΔN−34EcΔN=0.Equation (22) can be transformed into the following equation (23).(23)12ω0Ef¯−32N¯EKsinγ+14ECω0ΔEfcosγ+1−34ECΔNcosγ+1−ΔEfN¯+Ef¯ΔN=0.According to equations (15) and (16), the last two elements of equation (23) on the left side will be transformed into:(24)−ΔEfN¯+E¯fΔN=−2Ef1N2+2Ef2N1.In order to know the effect made by the gyration resistance coefficientsf1 and f2 of rotating shafts 1 and 2 for synchronization, we can set the last two elements of equation (23) on the left side at 0. And, both sides of equation (24) will be 0.(25)−2Ef1N2+2Ef2N1=0.According to the definition of equation (10), we can know that the sufficient condition for the establishment of equation (25) is that the discrepancy between the gyration resistance coefficients f1 and f2 is 0.In this way, equation (23) is simplified into the following equation (26).(26)12ω0Ef¯−32N¯EKsinγ+14ECω0ΔEfcosγ+1−34ECΔNcosγ+1=0,(27)LetA=12ω0Ef¯−32N¯EKB=14ω0Ef−34NEC.Then, the trigonometric solution to equation (26) is(28)cosγ=A2−B2A2+B2.So, the necessary criterion for the solution to synchronous equation (23) is(29)cosγ≤1.Consequently, the solutionγ exists.That is to say, the synchronous criterion is(30)A2−B2A2+B2=1−2B2A2+B2<1.Obviously, inequality (30) is always true. According to equation (25), we can know the meaning of equation (30) is that under the precondition that the gyration resistance coefficient, f1 and f2 are both 0, the gyration systems 1 and 2 will always be in sync regardless of structural parameters and gyration speed. So, reducing the gyration resistance coefficient of the gyration shaft is the most effective way to ensure the synchronous criterion. The conclusion is fully consistent with that of reference [3] obtained through electrohydraulic system-coupling modeling and simulation.The discussion above is a special case for synchronous criterion where the gyration resistance coefficient is 0. Then, the case was considered that gyration resistance coefficient is not 0.Equation (23) can be transformed into(31)Asinγ+Bcosγ+1+C=0,LetC=−ΔEfN¯+Ef¯ΔN,sinβ=BA2+B2.Then, the trigonometric solution to equation (31) is(32)sinγ+β=−B+CA2+B2,SetD=−B+CA2+B2.The necessary condition for synchronous criterion is(33)−B+CA2+B2≤1.IfC = 0, the sufficient condition for equation (25) is true, equation (33) can be transformed into(34)sinγ+β=−BA2+B2.Substitute equation (33) in equation (34), and then we get the solutions(35)γ=−2βorγ=−180∘.It shows that under the precondition that the gyration resistance coefficient is 0, the gyration systems 1 and 2 can keep constant sync with a phase difference nearπ or −2β, regardless of structural parameters, foundation soil stiffness, damping, and changes in the frequency. This conclusion is exactly the same as the one expressed by equation (30).The algebraic expressionD in equation (34) can be described as synchronous index. The absolute value of D can indicate the synchronous level for the strength of the vibration of the machine. The smaller D is (that is, the smaller the synchronous phase difference γ is), the easier synchronization is, and the lighter the control system “burden” is.


## 5. Sensitivity Analysis on Factors Affecting System Synchronization

According to the derivation of the system-coupling synchronization criterion, there are several factors that affect the coupling synchronism of the system, including mainly the difference in the structural parameters between the two rotation systems, the internal leakage of two hydraulic motors, and the parameters of the foundation soil as the load. According to the criteria, the higher the system synchronization is, the smaller the phase difference value between the two eccentric blocks is.According to the operation procedure of the orthogonal experiment design method [23], take the phase difference Δα between the two eccentric blocks of two rotation systems as an index, and take the difference between the rotary resistance coefficient of two motors Δf, the difference between the motor plunger value Δδ1, the synthetic stiffness of the foundation soil Ky, and the hydraulic system oil temperature T as factors affecting the index. The orthogonal experiment contains four factors, each containing five levels as follows (Table 1):Table 1
Orthogonal factor levels.
Factor
Δδ1 (mm)
T(°C)
Δf (%)
K
y (N/m)
1
1.0 × 10−6
45
2
1.4 × 108
2
1.5 × 10−6
50
4
2.0 × 108
3
2.0 × 10−6
55
6
2.6 × 108
4
2.5 × 10−6
60
8
3.2 × 108
5
3.0 × 10−6
65
10
3.8 × 108
Orthogonal table was selected as the orthogonal experimental design. The results are as follows (Table2).Table 2
Orthogonal results.
Program
Δδ1(mm)
T (°C)
Δf (%)
K
y (N/m)
Δα (rad)
1
1.0 × 10−6
45
2
1.4 × 108
2.538
2
1.0 × 10−6
50
4
2.0 × 108
2.057
3
1.0 × 10−6
55
6
2.6 × 108
2.946
4
1.0 × 10−6
60
8
3.2 × 108
3.340
5
1.0 × 10−6
65
10
3.8 × 108
3.516
6
1.5 × 10−6
45
4
2.6 × 108
3.083
7
1.5 × 10−6
50
6
3.2 × 108
2.973
8
1.5 × 10−6
55
8
3.8 × 108
4.418
9
1.5 × 10−6
60
10
1.4 × 108
4.503
10
1.5 × 10−6
65
2
2.0 × 108
3.112
11
2.0 × 10−6
45
6
3.8 × 108
2.833
12
2.0 × 10−6
50
8
1.4 × 108
2.059
13
2.0 × 10−6
55
10
2.0 × 108
3.131
14
2.0 × 10−6
60
2
2.6 × 108
3.109
15
2.0 × 10−6
65
4
3.2 × 108
2.964
16
2.5 × 10−6
45
8
2.0 × 108
2.905
17
2.5 × 10−6
50
10
2.6 × 108
2.526
18
2.5 × 10−6
55
2
3.2 × 108
2.829
19
2.5 × 10−6
60
4
3.8 × 108
3.169
20
2.5 × 10−6
65
6
1.4 × 108
3.154
21
3.0 × 10−6
45
2
3.8 × 108
3.290
22
3.0 × 10−6
50
4
2.0 × 108
3.077
23
3.0 × 10−6
55
6
2.6 × 108
4.385
24
3.0 × 10−6
60
8
3.2 × 108
3.017
25
3.0 × 10−6
65
10
1.4 × 108
3.106The range analysis of Table2 was performed, and the average phase difference Δα between two eccentric blocks under the influence of various factors and the sensitivity arrangement order of each factor were obtained, as shown in Table 3.Table 3
Range analysis results.
Factor
Δδ1
T
Δf
K
y
K
1j
1.150
1.121
1.390
0.854
K
2j
1.038
1.287
1.056
0.907
K
3j
1.064
1.106
1.148
0.615
K
4j
1.139
1.103
1.195
1.016
R
j
0.294
0.110
0.418
0.073
Sensitivity
2
3
1
4From the mean averageRj value of the above table, the difference between the rotary resistance coefficient of two motors Δf had the greatest impact on the system synchronization, accounting for 46.7% of the four influencing factors, followed by the difference between the motor plunger value Δδ1 and the hydraulic system oil temperature T, which accounted for 32.8% and 12.3% of the four influencing factors, respectively. The synthetic stiffness Ky of foundation soil had the weakest influence on the synchronization of the system, accounting for 8.2%, and the sensitivity was the lowest.Measures to improve synchronization are given as follows.It can be seen from the above study what sensitivity factors affect system synchronization, and correspondingly the following measures can be taken to improve synchronization:(1)
The difference between the rotary resistance coefficient of two motors Δf had the greatest influence on synchronization, accounting for 46.7% of the impact percentage. Therefore, reducing the difference between the rotary resistance coefficient of two motors Δf or minimizing the rotary resistance coefficient of two motors is the most effective way to achieve system synchronization.(2)
Secondly, as the difference between the motor plunger value Δδ1 had a great influence on the synchronization, accounting for 32.8% of the impact percentage, improving the manufacturing and processing quality of the two eccentric rotor systems and reducing the structural difference, especially the difference between the motor plunger value, can significantly improve the synchronization.(3)
Temperature also affected system synchronization, accounting for 12.3% of the impact; consequently, reducing the temperature difference between the two systems is also very important.(4)
The stiffness of foundation soil had the weakest influence on system synchronization, accounting for 8.2% of the impact percentage, which indicates that the influence of different foundation soils on the system synchronization is relatively weak, and the coupling synchronization vibration hammer can adapt to different construction environments.


## 6. Experiment

### 6.1. Change the Value ofmr to Decrease B in the Synchronization Criterion

Test flowchart is shown in Figure5.Figure 5
Test process.The main testing devices include ZZYY40 experimental hydraulic vibrating pile hammer, 2 sets of steel plate to display the angle of the eccentric rotary shaft, 1 set of vibration detector and its software, 1 set of high-speed cameras, etc.Figure6 shows the amplitude detection results under the given vibration pile condition, wherein mr = 11 kg·m and |D| = 0.65 were obtained after calculation. According to the theoretical criterion, the synchronization evaluation index is less than what satisfies the synchronization criterion. Figure 6 shows the experimental result that the amplitude was stable near 0.0015 m, so the synchronization was realized. The experiment results correspond to the theoretical analysis.Figure 6
y-t curve, withmr = 11 kg·m.Figure7 shows the amplitude detection results under the given vibration pile condition, wherein mr = 15 kg·m, the other conditions are the same as the test conditions in Figure 6, and |D| = 1.28 was obtained after calculation. According to the theoretical criterion, the synchronization evaluation index is more than 1, which fails to satisfy the synchronization criterion. Figure 7 shows the experimental result that the amplitude gradually decreased from the initial value of 0.00025 m. High-speed camera was used to shoot steel plate for displaying the angle of eccentric rotating axis, 10 shooting photos in 1 second. The results showed that the phase difference between two eccentric blocks changed without following the same pattern. The experiment results corresponded to the theoretical analysis.Figure 7
y-t curve, withmr = 15 kg·m.We also verified the synchronization criterion by adjusting the vibration frequency and the friction coefficient of the rotating system. All the experiments show that the synchronous evaluation model of the hydraulic-driven control system is correct, and the measures to improve the synchronization of the system are reasonable.


### 6.2. The Influence of the Difference between the Motor Plunger Value Δδ1 on System Synchronization

In this experiment, an optional matching method was used to match the Huade A2F80W2A2 slanting shaft plunger motor. The basic dimensions of the plunger and plunger holes are 18.9 mm. The dimensional tolerances of the plunger pairs selected from the two groups of motors are shown in Tables4 and 5, and the physical picture is shown in Figure 8.Table 4
The first group of dimensional tolerance table of the testing motor plunger pair.
First group
Motor A tolerance (mm)
Motor B tolerance (mm)
Plunger pair 1
Plunger
−0.011
−0.013
Hole
+0.010
+0.009
Plunger pair 2
Plunger
−0.015
−0.010
Hole
+0.006
+0.012
Plunger pair 3
Plunger
−0.011
−0.010
Hole
+0.012
+0.011
Plunger pair 4
Plunger
−0.012
−0.014
Hole
+0.013
+0.008
Plunger pair 5
Plunger
−0.013
−0.015
Hole
+0.009
+0.008
Plunger pair 6
Plunger
−0.014
−0.012
Hole
+0.007
+0.010
Plunger pair 7
Plunger
−0.010
−0.011
Hole
+0.015
+0.012Table 5
The second group of dimensional tolerance table of the testing motor plunger pair.
Second group
Motor A tolerance (mm)
Motor B tolerance (mm)
Plunger pair 1
Plunger
−0.015
−0.012
Hole
+0.005
+0.015
Plunger pair 2
Plunger
−0.010
−0.015
Hole
+0.010
+0.012
Plunger pair 3
Plunger
−0.011
−0.013
Hole
+0.010
+0.014
Plunger pair 4
Plunger
−0.013
−0.011
Hole
+0.015
+0.015
Plunger pair 5
Plunger
−0.010
−0.015
Hole
+0.010
+0.013
Plunger pair 6
Plunger
−0.013
−0.015
Hole
+0.007
+0.012
Plunger pair 7
Plunger
−0.014
−0.014
Hole
+0.006
+0.014Figure 8
The plunger pair of Huade A2F80W2A2.This experiment was based on the idea of single variable method, and only the difference in plunger pair clearance was changed. To make the other parameters as consistent as possible, the two groups of experiments were performed on the same test bench, and the system was cooled before starting the second group of experiments.Experimental results: when the gap difference of plunger pair of double hydraulic motors was relatively smaller (group 1), although the phase difference value fluctuated greatly, its peak value stayed within a certain range and did not diverge. However, when the gap difference between the plunger pair of two hydraulic motors was larger (group 2), the phase difference was rapidly divergent, and its value increased constantly. The validity of the synchronization criterion and the sensitive factors affecting synchronization were verified.


## 6.1. Change the Value ofmr to Decrease B in the Synchronization Criterion

Test flowchart is shown in Figure5.Figure 5
Test process.The main testing devices include ZZYY40 experimental hydraulic vibrating pile hammer, 2 sets of steel plate to display the angle of the eccentric rotary shaft, 1 set of vibration detector and its software, 1 set of high-speed cameras, etc.Figure6 shows the amplitude detection results under the given vibration pile condition, wherein mr = 11 kg·m and |D| = 0.65 were obtained after calculation. According to the theoretical criterion, the synchronization evaluation index is less than what satisfies the synchronization criterion. Figure 6 shows the experimental result that the amplitude was stable near 0.0015 m, so the synchronization was realized. The experiment results correspond to the theoretical analysis.Figure 6
y-t curve, withmr = 11 kg·m.Figure7 shows the amplitude detection results under the given vibration pile condition, wherein mr = 15 kg·m, the other conditions are the same as the test conditions in Figure 6, and |D| = 1.28 was obtained after calculation. According to the theoretical criterion, the synchronization evaluation index is more than 1, which fails to satisfy the synchronization criterion. Figure 7 shows the experimental result that the amplitude gradually decreased from the initial value of 0.00025 m. High-speed camera was used to shoot steel plate for displaying the angle of eccentric rotating axis, 10 shooting photos in 1 second. The results showed that the phase difference between two eccentric blocks changed without following the same pattern. The experiment results corresponded to the theoretical analysis.Figure 7
y-t curve, withmr = 15 kg·m.We also verified the synchronization criterion by adjusting the vibration frequency and the friction coefficient of the rotating system. All the experiments show that the synchronous evaluation model of the hydraulic-driven control system is correct, and the measures to improve the synchronization of the system are reasonable.


## 6.2. The Influence of the Difference between the Motor Plunger Value Δδ1 on System Synchronization

In this experiment, an optional matching method was used to match the Huade A2F80W2A2 slanting shaft plunger motor. The basic dimensions of the plunger and plunger holes are 18.9 mm. The dimensional tolerances of the plunger pairs selected from the two groups of motors are shown in Tables4 and 5, and the physical picture is shown in Figure 8.Table 4
The first group of dimensional tolerance table of the testing motor plunger pair.
First group
Motor A tolerance (mm)
Motor B tolerance (mm)
Plunger pair 1
Plunger
−0.011
−0.013
Hole
+0.010
+0.009
Plunger pair 2
Plunger
−0.015
−0.010
Hole
+0.006
+0.012
Plunger pair 3
Plunger
−0.011
−0.010
Hole
+0.012
+0.011
Plunger pair 4
Plunger
−0.012
−0.014
Hole
+0.013
+0.008
Plunger pair 5
Plunger
−0.013
−0.015
Hole
+0.009
+0.008
Plunger pair 6
Plunger
−0.014
−0.012
Hole
+0.007
+0.010
Plunger pair 7
Plunger
−0.010
−0.011
Hole
+0.015
+0.012Table 5
The second group of dimensional tolerance table of the testing motor plunger pair.
Second group
Motor A tolerance (mm)
Motor B tolerance (mm)
Plunger pair 1
Plunger
−0.015
−0.012
Hole
+0.005
+0.015
Plunger pair 2
Plunger
−0.010
−0.015
Hole
+0.010
+0.012
Plunger pair 3
Plunger
−0.011
−0.013
Hole
+0.010
+0.014
Plunger pair 4
Plunger
−0.013
−0.011
Hole
+0.015
+0.015
Plunger pair 5
Plunger
−0.010
−0.015
Hole
+0.010
+0.013
Plunger pair 6
Plunger
−0.013
−0.015
Hole
+0.007
+0.012
Plunger pair 7
Plunger
−0.014
−0.014
Hole
+0.006
+0.014Figure 8
The plunger pair of Huade A2F80W2A2.This experiment was based on the idea of single variable method, and only the difference in plunger pair clearance was changed. To make the other parameters as consistent as possible, the two groups of experiments were performed on the same test bench, and the system was cooled before starting the second group of experiments.Experimental results: when the gap difference of plunger pair of double hydraulic motors was relatively smaller (group 1), although the phase difference value fluctuated greatly, its peak value stayed within a certain range and did not diverge. However, when the gap difference between the plunger pair of two hydraulic motors was larger (group 2), the phase difference was rapidly divergent, and its value increased constantly. The validity of the synchronization criterion and the sensitive factors affecting synchronization were verified.


## 7. Conclusions

The coupling items (α1, α2, Δα1, and Δα2) in equations (1) and (2) are symmetrically distributed in the dynamic formula, and according to the equations (13) and (14), the microangular velocity solution indicate that the minor variation of rotational motion in the two rotating systems is mutually limited.The absolute value of synchronous indexD, which is smaller than 1, can indicate the synchronous level for the strength of the vibration of machine. By analyzing the synchronous index D, it was found that the resistance coefficient difference and the difference in motor piston clearance have the greatest influence on the synchronous motion between the two motors. Of all the factors discussed in this paper, the influence of the resistance coefficient difference on system motion stability is the greatest, accounting for 46.7%, and the influence of the difference in motion pair structure (e.g. motor piston clearance) is the second, accounting for 32.8%.For motors with a displacement of 80 ml/r, the condition of self-synchronization is that the difference in piston clearance between the two motors is equal to or smaller than 6μm.



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*Source: 6086874-2019-01-20.xml*