# Theoretical and Experimental Study on Electromechanical Coupling Properties of Multihammer Synchronous Vibration System

Authors: Xin Lai; Wanjun Xie
Journal: Shock and Vibration
 (2016)

---

## Abstract

Industrial simulation of real external load using multiple exciting points or increasing exciting force by synchronizing multiple exciting forces requires multiple vibration hammers to be coordinated and work together. Multihammer vibration system which consists of several hammers is a complex electromechanical system with complex electromechanical coupling. In this paper, electromechanical coupling properties of such a multihammer vibration system were studied in detail using theoretical derivation, numerical simulation, and experiment. A kinetic model of multihammer synchronous vibration system was established, and approximate expressions for electromechanical coupling strength were solved using a small parameter periodic averaging method. Basic coupling rules and reasons were obtained. Self-synchronization and frequency hopping phenomenon were also analyzed. Subsequently, numerical simulations were carried out and electromechanical coupling process was obtained for different parameters. Simulation results verify correctness of the proposed model and results. Finally, experiments were carried out, self-synchronization and frequency hopping phenomenon were both observed, and results agree well with theoretical deduction and simulation results. These results provide theoretical foundations for multihammer synchronous vibration system and its synchronous control.


---

## Body

## 1. Introduction

The vibratory hammer driven by electromotor is widely used in construction engineering. The use of a single hammer is restricted by the limited power of electromotor in large pile foundation projects. Therefore, multihammer synchronous vibration system using multiple vibration hammers coordinated to work together was designed and applied to multiple projects in recent years [1]. One such application was demonstrated by Japanese engineers during the construction of Kansai international airport. Here, eight 150 kW vibratory pile hammers with linkage shafts in series were used to vibrate and sink a large steel cylinder 23 m in diameter. Another such application was demonstrated for construction applications by a Chinese team, where four hydraulic pile hammers linked with bevel gears were used to sink a cylinder pile 13.5 m in diameter. The American Pile Driving Equipment Corp. used eight linked pile hammers while working on the man-made island project (Hong Kong-Zhuhai-Macao Bridge, China) [2]. For current engineering applications, the shaft coupling has been applied to connect the rotating shaft of each vibratory hammer. This allows both phase and velocity of the vibratory hammers to be synchronized. However, a disadvantage of this method is the use of complex linkage structures, which can be easily damaged; meanwhile, the number of vibration hammers is also not easy to expand. One solution to this problem was proposed by PVE Corp. (Netherlands), where an electric control linkage method with phase difference was electrically measured to allow real-time correction done by software. Here, multiple hammers were placed on a pile in a parallel configuration, phase difference between eccentric blocks of pile hammers was measured, and velocity and phase synchronization was achieved using a control system, allowing the maximum energy and excited force synthesis [3]. However, there are no subsequent reports and engineering applications of this method so far. Bingham et al. removed the mechanical gear assembly between the eccentric masses and implemented a control system to realize synchronization of different eccentric masses [4]. Electromechanical coupling was discovered in this system; however, it was regarded as a disturbance signal in the design of control algorithm.In order to perform industrial simulation of a real external load, multiple excitation points need to be set. For example, wind turbine blade loading multipoint excitation system uses a multihammer to excite the wind turbine blade [5, 6]. This allows the load approximation to be closer to values in practical scenarios. Similar to the previous case, a strong electromechanical coupling was seen for this system.Electromechanical coupling properties of vibrating systems have been extensively studied and reported in literature since the discovery of the self-synchronization phenomenon in 1665 by Huygens [7]. Blekhman et al. [8, 9] proposed a definition to generalize mechanical synchronization and the basic synchronous vibration theory for a double vibration hammer. Inoue et al. [10] discovered three-octave synchronization for double motor driving vibrators. Wen et al. [11–13] established synchronization conditions for two vibratory hammers. In these studies, a majority of the reported research has focused on mechanical system such as the vibrating feeder, vibrating conveyer, and vibrating screen. In addition, computer simulation was frequently used. However, a systematic study to evaluate the electromechanical coupling properties of a multihammer system has not been reported in literature. In order to fully understand the electromechanical coupling of this system and then use and control it, it is critical to evaluate the electromechanical coupling characteristics using both theoretical and experimental methods.Taking two-hammer synchronous vibration system, for example, we conducted this study to derive and validate the electromechanical coupling properties of this system. The paper is organized as follows. The dynamics equation and mathematical model of two-hammer synchronous vibration system are set up in Section2. Using an approximate analysis for the coupling mechanism, an expression along with influence factors for the coupling strength is determined in Section 3. Next, self-synchronization and frequency hopping phenomenon are analyzed in Section 4. Then numerical simulation of the electromechanical coupling process is carried out in Section 5. Experiments are carried out to verify the correctness of theoretical derivation and simulation result in Section 6. Finally, conclusions are provided in Section 7.


## 2. Mathematical Model

### 2.1. Multihammer Synchronous Vibration System

The working principle of vibratory hammer can be described as follows: two groups of eccentric block with equal eccentric torque are droved by electromotor; in this case, each pair of eccentric blocks is reversed at the same angular velocity. Hence, horizontal centrifugal force is canceled out, and vertical centrifugal force is added, as shown in Figure1. Differential equations for the motion and steady response of vibratory hammer can be expressed as(1)
m
y
¨
t
+
c
y
˙
t
+
k
y
t
=
N
ω
2
r
c
m
0
cos
⁡
θ
2
cos
⁡
ω
t
+
θ
2
,
Y
t
=
N
ω
2
r
c
m
0
cos
⁡
θ
/
2
k
-
m
ω
2
2
+
C
2
ω
2
1
/
2
cos
⁡
ω
t
-
ϕ
,where Y
(
t
) is steady-state displacement response function, N is the number of eccentric masses, ω is rotational angular velocity of eccentric mass, r
c is eccentricity of eccentric mass, m
0 is the quality of the single chip eccentric mass, c is damping coefficient, k is stiffness coefficient, and θ is angle between a pair of eccentric blocks.Figure 1
Structural model of vibratory hammer.For a single vibratory hammer, vibration force is directly related toθ and ω. Vibration frequency can be varied by changing the value of ω, and amplitude can be controlled by varying θ; this is the principle of continuous frequency and amplitude modulation vibratory hammer. Since vibration force that can be generated by a single vibratory hammer is limited, an obvious extension to increase force for engineering application is achieved by connecting multiple vibrating hammers together. The working principle of such a multihammer synchronous vibration system can be described as follows: multiple hammers concurrently shock the same object; eccentric masses of all hammers rotate at the same speed (it is defined as frequency synchronization or speed synchronization) and the same eccentric mass position (it is defined as phase synchronization); frequency and phase synchronization can be realized through adjusting motor speed of the vibratory hammer. When all the eccentric masses achieve both speed and phase synchronization, this is referred to as a multihammer synchronous vibration system with maximum resultant exciting force. Figure 2 shows a schematic of multihammer synchronous vibration pile sinking system. In this design, multiple hammers are used to shock the same pile, with the aim of making multiple “small” hammers into one “big” hammer.Figure 2
Multihammer synchronous vibration system.


### 2.2. Mathematical Modeling

In order to simplify modeling and solving, then obtain the basic electromechanical coupling rules; two-hammer synchronous vibration system is considered here. The two-hammer synchronous vibration system has two pile hammers fixed into a pile so that the “hammer-pile-soil” constitutes a complex system. During the piling process, stiffness of soil is much smaller than that of the pile. Based on the theory of single degree of freedom proposed by Blekhman [14], two assumptions are taken into account for modeling: first, soil is regarded as an elastomer, while the hammers and pile are considered as absolutely homogeneous rigid bodies. Secondly, damping force and elastic force of the soil against the pile when the pile vibrates are a linear function of pile movement speed and displacement, respectively. Based on the above two assumptions, dynamic model of two-hammer synchronous vibration system is illustrated in Figure 3.Figure 3
Model of two-pile-hammer synchronous vibration system.In Figure3, o
x
y
z is absolute coordinate system, o
′
x
′
y
′
z
′ is moving coordinate system, φ is angle between o
x
y
z and o
′
x
′
y
′
z
′, o
i is midpoint of rotating center of eccentric masses of hammer, l
i
=
o
o
i, β
=
∠
o
i
o
x, m
h is mass of the hammer and pile (not including eccentric block), m
i is mass of the eccentric block, θ
i is rotation angle of rotor, ω
i is angular velocity of eccentric mass, ω
i
=
θ
˙
i, and r
i is eccentric distance of the eccentric block (
i
=
1,2
).Choosingx, y, φ, θ
i as generalized coordinates, the kinetic energy, potential energy, and generalized force of vibration system can be written as(2)
T
=
1
2
J
h
φ
˙
2
+
∑
i
=
1
2
J
i
θ
i
2
+
1
2
m
h
x
˙
-
φ
˙
l
0
sin
⁡
β
0
2
+
y
˙
+
φ
˙
l
0
cos
⁡
β
0
2
+
∑
i
=
1
2
m
i
x
˙
+
φ
˙
l
i
sin
⁡
β
i
2
+
y
˙
-
φ
˙
l
i
cos
⁡
β
i
+
θ
i
r
i
cos
⁡
θ
i
2
,
V
=
1
2
k
x
x
2
+
1
2
k
y
y
2
+
1
2
k
φ
φ
2
,
Q
i
=
-
c
x
x
˙
-
c
y
y
˙
-
c
φ
φ
˙
T
em
1
-
c
1
θ
˙
1
T
em
2
-
c
2
θ
˙
2
T
,where T is kinetic energy, V is potential energy, Q
i is generalized force, J
h is moment of inertia of hammers and pile, J
i is moment of inertia of the rotor, c
i is the damping constant of rotor of motor, T
e
m
i is electromagnetic torque of the motor (
i
=
1,2
), k
x, k
y, and k
φ are the stiffness coefficients of soil in the direction of x, y, and φ, respectively, and c
x, c
y, and c
φ are damping coefficients of the soil in the direction of x, y, and φ, respectively.Hence, substituting (2) into Lagrange equations, the dynamic equations of two-hammer vibration system can be established as(3)
m
h
+
2
∑
i
=
1
2
m
i
x
¨
+
c
x
x
˙
+
k
x
x
=
0
,
m
h
+
2
∑
i
=
1
2
m
i
y
¨
+
c
y
y
˙
+
k
y
y
=
2
∑
i
=
1
2
m
i
r
i
θ
˙
i
2
sin
⁡
θ
i
-
θ
¨
i
cos
⁡
θ
i
,
∑
i
=
1
2
m
i
l
i
2
+
J
h
φ
¨
+
c
φ
φ
˙
+
k
φ
φ
=
2
∑
i
=
1
2
m
i
l
i
r
i
cos
⁡
β
i
θ
¨
i
cos
⁡
θ
i
-
θ
˙
i
2
sin
⁡
θ
i
,
J
i
+
m
i
r
i
2
+
θ
¨
i
+
c
i
θ
˙
i
=
T
em
i
-
m
i
r
i
sin
⁡
θ
i
θ
˙
i
-
1
y
˙
-
φ
˙
l
i
cos
⁡
β
i
-
θ
˙
i
2
r
i
cos
⁡
θ
i
-
m
i
r
i
cos
⁡
θ
i
y
¨
-
φ
¨
l
i
cos
⁡
β
i
+
θ
¨
i
r
i
cos
⁡
θ
i
.


### 2.3. Mathematical Model for Three-Phase Asynchronous Motor

The electric vibratory hammer uses a three-phase asynchronous motor. Therefore, according to the literature [15], the mathematical model of motor can be expressed as(4)
I
˙
d
s
I
˙
q
s
I
˙
d
r
I
˙
q
r
=
1
L
m
2
-
L
s
L
r
L
s
R
s
ω
s
L
s
L
r
-
L
m
2
+
ω
r
L
m
2
-
L
m
R
r
-
ω
r
L
m
L
r
ω
s
L
s
L
r
-
L
m
2
+
ω
r
L
m
2
L
r
R
s
ω
r
L
r
L
m
-
L
m
R
r
-
L
m
R
s
ω
r
L
m
L
s
L
s
R
r
ω
s
L
r
L
s
-
L
m
2
-
L
s
L
m
ω
r
-
L
m
R
s
ω
s
L
r
L
s
-
L
m
2
R
r
L
s
I
d
s
I
q
s
I
d
r
I
q
r
+
-
L
r
0
-
L
m
0
0
-
L
r
0
L
m
L
m
0
-
L
s
0
0
L
m
0
-
L
s
U
d
s
U
q
s
U
d
r
U
q
r
,
ω
˙
r
=
n
p
2
J
0
L
m
I
q
1
I
d
2
-
I
q
2
I
d
1
-
D
J
0
ω
r
-
n
p
J
0
T
L
,where U
d
s, U
q
s, U
d
r, and U
q
r are stator and rotor voltage in a two-phase synchronous rotating coordinate system, I
d
s, I
q
s, I
d
r, and I
q
r are stator and rotor current in a two-phase synchronous rotating coordinate system, R
s, R
r are resistance of the stator and rotor, L
s, L
r are inductance of the stator and rotor, L
m is mutual induction of the stator and rotor, ω
s is synchronous rotation angular velocity, ω
r is angular velocity of the rotor, D is coefficient of drag torque, T
L is load torque of the induction motors, n
p is pole pairs of the motor, and J
0 is moment of inertia of the induction motors.Hence, the mathematical model for the electromechanical coupling of the two-hammer synchronous vibration system can be expressed as (3)-(4); it is clear that the parameters between the vibration system and the motor system are mutually coupled. It represents the mathematical coupling relation in the “hammer-pile-soil” system during the process of pile sinking.


## 2.1. Multihammer Synchronous Vibration System

The working principle of vibratory hammer can be described as follows: two groups of eccentric block with equal eccentric torque are droved by electromotor; in this case, each pair of eccentric blocks is reversed at the same angular velocity. Hence, horizontal centrifugal force is canceled out, and vertical centrifugal force is added, as shown in Figure1. Differential equations for the motion and steady response of vibratory hammer can be expressed as(1)
m
y
¨
t
+
c
y
˙
t
+
k
y
t
=
N
ω
2
r
c
m
0
cos
⁡
θ
2
cos
⁡
ω
t
+
θ
2
,
Y
t
=
N
ω
2
r
c
m
0
cos
⁡
θ
/
2
k
-
m
ω
2
2
+
C
2
ω
2
1
/
2
cos
⁡
ω
t
-
ϕ
,where Y
(
t
) is steady-state displacement response function, N is the number of eccentric masses, ω is rotational angular velocity of eccentric mass, r
c is eccentricity of eccentric mass, m
0 is the quality of the single chip eccentric mass, c is damping coefficient, k is stiffness coefficient, and θ is angle between a pair of eccentric blocks.Figure 1
Structural model of vibratory hammer.For a single vibratory hammer, vibration force is directly related toθ and ω. Vibration frequency can be varied by changing the value of ω, and amplitude can be controlled by varying θ; this is the principle of continuous frequency and amplitude modulation vibratory hammer. Since vibration force that can be generated by a single vibratory hammer is limited, an obvious extension to increase force for engineering application is achieved by connecting multiple vibrating hammers together. The working principle of such a multihammer synchronous vibration system can be described as follows: multiple hammers concurrently shock the same object; eccentric masses of all hammers rotate at the same speed (it is defined as frequency synchronization or speed synchronization) and the same eccentric mass position (it is defined as phase synchronization); frequency and phase synchronization can be realized through adjusting motor speed of the vibratory hammer. When all the eccentric masses achieve both speed and phase synchronization, this is referred to as a multihammer synchronous vibration system with maximum resultant exciting force. Figure 2 shows a schematic of multihammer synchronous vibration pile sinking system. In this design, multiple hammers are used to shock the same pile, with the aim of making multiple “small” hammers into one “big” hammer.Figure 2
Multihammer synchronous vibration system.


## 2.2. Mathematical Modeling

In order to simplify modeling and solving, then obtain the basic electromechanical coupling rules; two-hammer synchronous vibration system is considered here. The two-hammer synchronous vibration system has two pile hammers fixed into a pile so that the “hammer-pile-soil” constitutes a complex system. During the piling process, stiffness of soil is much smaller than that of the pile. Based on the theory of single degree of freedom proposed by Blekhman [14], two assumptions are taken into account for modeling: first, soil is regarded as an elastomer, while the hammers and pile are considered as absolutely homogeneous rigid bodies. Secondly, damping force and elastic force of the soil against the pile when the pile vibrates are a linear function of pile movement speed and displacement, respectively. Based on the above two assumptions, dynamic model of two-hammer synchronous vibration system is illustrated in Figure 3.Figure 3
Model of two-pile-hammer synchronous vibration system.In Figure3, o
x
y
z is absolute coordinate system, o
′
x
′
y
′
z
′ is moving coordinate system, φ is angle between o
x
y
z and o
′
x
′
y
′
z
′, o
i is midpoint of rotating center of eccentric masses of hammer, l
i
=
o
o
i, β
=
∠
o
i
o
x, m
h is mass of the hammer and pile (not including eccentric block), m
i is mass of the eccentric block, θ
i is rotation angle of rotor, ω
i is angular velocity of eccentric mass, ω
i
=
θ
˙
i, and r
i is eccentric distance of the eccentric block (
i
=
1,2
).Choosingx, y, φ, θ
i as generalized coordinates, the kinetic energy, potential energy, and generalized force of vibration system can be written as(2)
T
=
1
2
J
h
φ
˙
2
+
∑
i
=
1
2
J
i
θ
i
2
+
1
2
m
h
x
˙
-
φ
˙
l
0
sin
⁡
β
0
2
+
y
˙
+
φ
˙
l
0
cos
⁡
β
0
2
+
∑
i
=
1
2
m
i
x
˙
+
φ
˙
l
i
sin
⁡
β
i
2
+
y
˙
-
φ
˙
l
i
cos
⁡
β
i
+
θ
i
r
i
cos
⁡
θ
i
2
,
V
=
1
2
k
x
x
2
+
1
2
k
y
y
2
+
1
2
k
φ
φ
2
,
Q
i
=
-
c
x
x
˙
-
c
y
y
˙
-
c
φ
φ
˙
T
em
1
-
c
1
θ
˙
1
T
em
2
-
c
2
θ
˙
2
T
,where T is kinetic energy, V is potential energy, Q
i is generalized force, J
h is moment of inertia of hammers and pile, J
i is moment of inertia of the rotor, c
i is the damping constant of rotor of motor, T
e
m
i is electromagnetic torque of the motor (
i
=
1,2
), k
x, k
y, and k
φ are the stiffness coefficients of soil in the direction of x, y, and φ, respectively, and c
x, c
y, and c
φ are damping coefficients of the soil in the direction of x, y, and φ, respectively.Hence, substituting (2) into Lagrange equations, the dynamic equations of two-hammer vibration system can be established as(3)
m
h
+
2
∑
i
=
1
2
m
i
x
¨
+
c
x
x
˙
+
k
x
x
=
0
,
m
h
+
2
∑
i
=
1
2
m
i
y
¨
+
c
y
y
˙
+
k
y
y
=
2
∑
i
=
1
2
m
i
r
i
θ
˙
i
2
sin
⁡
θ
i
-
θ
¨
i
cos
⁡
θ
i
,
∑
i
=
1
2
m
i
l
i
2
+
J
h
φ
¨
+
c
φ
φ
˙
+
k
φ
φ
=
2
∑
i
=
1
2
m
i
l
i
r
i
cos
⁡
β
i
θ
¨
i
cos
⁡
θ
i
-
θ
˙
i
2
sin
⁡
θ
i
,
J
i
+
m
i
r
i
2
+
θ
¨
i
+
c
i
θ
˙
i
=
T
em
i
-
m
i
r
i
sin
⁡
θ
i
θ
˙
i
-
1
y
˙
-
φ
˙
l
i
cos
⁡
β
i
-
θ
˙
i
2
r
i
cos
⁡
θ
i
-
m
i
r
i
cos
⁡
θ
i
y
¨
-
φ
¨
l
i
cos
⁡
β
i
+
θ
¨
i
r
i
cos
⁡
θ
i
.


## 2.3. Mathematical Model for Three-Phase Asynchronous Motor

The electric vibratory hammer uses a three-phase asynchronous motor. Therefore, according to the literature [15], the mathematical model of motor can be expressed as(4)
I
˙
d
s
I
˙
q
s
I
˙
d
r
I
˙
q
r
=
1
L
m
2
-
L
s
L
r
L
s
R
s
ω
s
L
s
L
r
-
L
m
2
+
ω
r
L
m
2
-
L
m
R
r
-
ω
r
L
m
L
r
ω
s
L
s
L
r
-
L
m
2
+
ω
r
L
m
2
L
r
R
s
ω
r
L
r
L
m
-
L
m
R
r
-
L
m
R
s
ω
r
L
m
L
s
L
s
R
r
ω
s
L
r
L
s
-
L
m
2
-
L
s
L
m
ω
r
-
L
m
R
s
ω
s
L
r
L
s
-
L
m
2
R
r
L
s
I
d
s
I
q
s
I
d
r
I
q
r
+
-
L
r
0
-
L
m
0
0
-
L
r
0
L
m
L
m
0
-
L
s
0
0
L
m
0
-
L
s
U
d
s
U
q
s
U
d
r
U
q
r
,
ω
˙
r
=
n
p
2
J
0
L
m
I
q
1
I
d
2
-
I
q
2
I
d
1
-
D
J
0
ω
r
-
n
p
J
0
T
L
,where U
d
s, U
q
s, U
d
r, and U
q
r are stator and rotor voltage in a two-phase synchronous rotating coordinate system, I
d
s, I
q
s, I
d
r, and I
q
r are stator and rotor current in a two-phase synchronous rotating coordinate system, R
s, R
r are resistance of the stator and rotor, L
s, L
r are inductance of the stator and rotor, L
m is mutual induction of the stator and rotor, ω
s is synchronous rotation angular velocity, ω
r is angular velocity of the rotor, D is coefficient of drag torque, T
L is load torque of the induction motors, n
p is pole pairs of the motor, and J
0 is moment of inertia of the induction motors.Hence, the mathematical model for the electromechanical coupling of the two-hammer synchronous vibration system can be expressed as (3)-(4); it is clear that the parameters between the vibration system and the motor system are mutually coupled. It represents the mathematical coupling relation in the “hammer-pile-soil” system during the process of pile sinking.


## 3. Approximate Analysis of Electromechanical Coupling Strength

Equations (3)-(4) show that it is quite difficult to obtain accurate analytical solution. In this paper, load torque of driving motor is analyzed according to approximate solving (3); then electromechanical coupling characteristics are analyzed according to the working characteristics of motor. In order to simplify the problem of solving (3), moment balance of two hammers is analyzed under ideal conditions, where two hammers are symmetrically installed, and their mechanical structures and geometric dimensions are identical; that is, m
1
=
m
2
=
m
0, r
1
=
r
2
=
r
0, l
1
=
l
2
=
l
0, and β
1
=
β
2
=
β
0. In this case, the method of parameter period averaging is adopted to determine an approximate solution for the system of equations. The torque change rule of the two hammer motors was analyzed to reveal the electromechanical coupling property of the system.Suppose that the average rotating angle of rotor 1 and rotor 2 isθ
∗, and average rotating speed of rotors is ω
0; the phase difference between rotors is 2
α; that is, 2
α
=
θ
1
-
θ
2. Hence, rotating angles of rotors can therefore be expressed as(5)
θ
1
=
θ
∗
+
α
,
θ
2
=
θ
∗
-
α
.Suppose that the fluctuation coefficient forω
0 is ε
1, while that of instantaneous phase difference relative to ω
0 is ε
2, and ε
1 and ε
2 are functions of time. If the changes of ε
1 and ε
2 tend to zero (i.e., ε
˙
1
=
0, ε
˙
2
=
0), the two rotors can operate synchronously. Thus,(6)
θ
˙
∗
=
ω
0
1
+
ε
1
,
α
˙
=
ω
0
ε
2
.From (5) and (6), instantaneous rotation velocity of two rotors can be written as (7)
θ
˙
1
=
ω
0
1
+
ε
1
+
ε
2
,
θ
˙
2
=
ω
0
1
+
ε
1
-
ε
2
.From (7), instantaneous accelerated velocity of two rotors can be written as(8)
θ
¨
1
=
ω
0
ε
˙
1
+
ε
˙
2
,
θ
¨
2
=
ω
0
ε
˙
1
-
ε
˙
2
.Since speed fluctuation is much less thanω
0, the values of ε
1 and ε
2 are far less than one. Additionally, ε
1 and ε
2 are slowly varying, so ε
˙
1 and ε
˙
2 are very small. According to (8), θ
¨
1 and θ
¨
2 are also very small, so θ
¨
i can be ignored in the second and third equations of (3). In this case, substituting (5) and (8) into (3), the following equations can be obtained:(9)
y
¨
+
c
y
M
y
˙
+
ω
y
2
y
=
ω
0
2
p
1
+
ε
1
+
ε
2
2
sin
⁡
θ
∗
+
α
+
1
+
ε
1
-
ε
2
2
sin
⁡
θ
∗
-
α
,
φ
¨
+
c
φ
J
φ
˙
+
ω
φ
2
φ
=
-
ω
0
2
q
cos
⁡
β
0
1
+
ε
1
+
ε
2
2
sin
⁡
θ
∗
+
α
+
1
+
ε
1
-
ε
2
2
sin
⁡
θ
∗
-
α
,where M
=
m
h
+
4
m
0, J
=
2
m
0
l
0
2
+
J
h, p
=
2
m
0
r
0
/
M, q
=
2
m
0
l
0
r
0
/
J, ω
y
=
k
y
/
M, and ω
φ
=
k
y
/
J.According to the superposition principle of linear equations, solution of (9) can be expressed as(10)
y
=
A
y
sin
⁡
θ
∗
+
α
+
γ
y
+
A
y
sin
⁡
θ
∗
-
α
+
γ
y
,
φ
=
A
φ
sin
⁡
θ
∗
+
α
+
γ
φ
+
A
φ
sin
⁡
θ
∗
-
α
+
γ
φ
,where A
y
=
p
/
ω
y
/
ω
0
2
-
1
2
+
c
y
/
M
2, A
φ
=
-
q
/
ω
φ
/
ω
0
2
-
1
2
+
c
φ
/
M
2, γ
y
=
arctan
⁡
c
y
ω
0
/
k
y
-
M
ω
0
2,  and γ
φ
=
arctan
⁡
c
φ
ω
0
/
k
φ
-
M
ω
0
2.Furthermore, the fourth term of (3) can be rewritten as(11)
J
θ
i
θ
¨
i
+
f
θ
i
,
y
˙
,
φ
˙
θ
˙
i
=
T
e
m
i
+
T
f
i
+
T
n
i
,where J
(
θ
i
)
=
J
i
+
m
0
r
0
2
c
o
s
2
θ
i, f
(
θ
i
,
y
˙
,
φ
˙
)
=
2
m
0
r
0
y
˙
-
φ
˙
l
0
cos
⁡
β
0
sin
⁡
θ
i
-
θ
˙
i
m
0
r
0
2
sin
⁡
2
θ
i, T
f
i
=
2
m
0
r
0
y
˙
-
φ
˙
l
0
cos
⁡
β
0
sin
⁡
θ
i, and T
n
i
=
2
m
0
r
0
l
0
cos
⁡
β
0
φ
¨
-
y
¨
cos
⁡
θ
i, i
=
1
,
2.Here,T
f
i is closely related to y
˙ and φ
˙; we define it as the vibration resistance moment. T
n
i is closely related to y
¨ and φ
¨, so we define it as vibration inertia moment. The cause of electromechanical coupling is interaction and mutual influence between the vibration state of vibration system and the speed of the motors, so T
f
i and T
n
i can describe the electromechanical coupling relationship of two-hammer synchronous vibration system.From (10), the first derivatives of y and φ can be written as(12)
y
˙
=
A
y
ω
0
cos
⁡
θ
1
cos
⁡
γ
y
1
+
ε
1
+
ε
2
+
cos
⁡
θ
2
cos
⁡
γ
y
1
+
ε
1
-
ε
2
-
sin
⁡
θ
1
+
sin
⁡
θ
2
sin
⁡
γ
y
,
φ
˙
=
A
φ
ω
0
cos
⁡
θ
1
cos
⁡
γ
φ
1
+
ε
1
+
ε
2
+
cos
⁡
θ
2
cos
⁡
γ
φ
1
+
ε
1
-
ε
2
-
sin
⁡
θ
1
+
sin
⁡
θ
2
sin
⁡
γ
φ
.Therefore, the second derivatives ofy and φ can be obtained as follows: (13)
y
¨
=
A
y
ω
0
cos
⁡
θ
1
cos
⁡
γ
y
ε
˙
1
+
ε
˙
2
-
sin
⁡
θ
1
cos
⁡
γ
y
1
+
ε
1
+
ε
2
-
sin
⁡
γ
y
cos
⁡
θ
1
+
cos
⁡
θ
2
+
cos
⁡
θ
2
cos
⁡
γ
y
ε
˙
1
-
ε
˙
2
-
sin
⁡
θ
2
cos
⁡
γ
y
1
+
ε
1
-
ε
2
,
φ
¨
=
A
φ
ω
0
cos
⁡
θ
1
cos
⁡
γ
φ
ε
˙
1
+
ε
˙
2
-
sin
⁡
θ
1
cos
⁡
γ
φ
1
+
ε
1
+
ε
2
-
sin
⁡
γ
φ
cos
⁡
θ
1
+
cos
⁡
θ
2
+
cos
⁡
θ
2
cos
⁡
γ
φ
ε
˙
1
-
ε
˙
2
-
sin
⁡
θ
2
cos
⁡
γ
φ
1
+
ε
1
-
ε
2
.Now, average values ofT
f
i and T
n
i within a vibration period can be expressed as(14)
T
-
f
i
=
1
2
π
∫
0
2
π
T
f
i
d
θ
=
B
0
A
y
ε
1
+
-
1
i
ε
2
sin
⁡
2
α
+
γ
y
+
2
+
ε
1
+
-
1
i
ε
2
sin
⁡
2
α
-
γ
y
-
2
sin
⁡
γ
y
-
B
1
A
φ
ε
1
+
-
1
i
ε
2
sin
⁡
2
α
+
γ
φ
+
2
+
ε
1
+
-
1
i
ε
2
sin
⁡
2
α
-
γ
φ
-
2
sin
⁡
γ
φ
,
T
-
n
i
=
1
2
π
∫
0
2
π
T
n
i
d
θ
=
B
1
A
φ
ε
1
+
-
1
i
ε
2
sin
⁡
2
α
+
γ
φ
+
2
+
ε
1
+
-
1
i
ε
2
sin
⁡
2
α
-
γ
φ
+
ε
˙
1
+
-
1
i
ε
˙
2
cos
⁡
2
α
+
γ
φ
+
cos
⁡
2
α
-
γ
φ
+
2
ε
˙
1
-
-
1
i
ε
˙
2
cos
⁡
γ
φ
-
2
sin
⁡
γ
φ
-
B
0
A
y
ε
1
+
-
1
i
ε
2
sin
⁡
2
α
+
γ
y
+
2
+
ε
1
+
-
1
i
ε
2
sin
⁡
2
α
-
γ
y
+
ε
˙
1
+
-
1
i
ε
˙
2
cos
⁡
2
α
+
γ
y
+
cos
⁡
2
α
-
γ
y
+
2
ε
˙
1
-
-
1
i
ε
˙
2
cos
⁡
γ
y
-
2
sin
⁡
γ
y
,where B
0
=
m
0
ω
0
r
0
/
2, B
1
=
m
0
ω
0
r
0
l
0
cos
⁡
β
0
/
2, T
-
f
i is average vibration resistance moment, and T
-
n
i is average vibration inertia moment, i
=
1,2.According to above analysis of the cause of electromechanical coupling, from (14), average coupling torque of Motor 1 and Motor 2 within one vibration period can be obtained as(15)
T
-
C
1
=
T
-
f
1
+
T
-
n
1
=
-
2
λ
ε
˙
1
cos
⁡
α
+
ε
˙
2
sin
⁡
2
α
,
T
-
C
2
=
T
-
f
2
+
T
-
n
2
=
-
2
λ
ε
˙
1
cos
⁡
α
-
ε
˙
2
sin
⁡
2
α
,where λ
=
B
0
A
y
cos
⁡
γ
y
-
B
1
A
φ
cos
⁡
γ
φ, T
-
C
1 is defined as average coupling torque of Motor 1, and T
-
C
2 is defined as average coupling torque of Motor 2.From (15), absolute value |
Δ
T
-
C
| of coupling torque difference of the two motors can be expressed as(16)
Δ
T
-
C
=
T
-
C
1
-
T
-
C
2
=
4
λ
ε
˙
2
sin
⁡
2
α
.According to working characteristics of three-phase asynchronous motor, with increase of load torque, the motor slip ratio increases, and rotation speed decreases. Moreover, the variation of motor speed will directly affect the vibration state of two-hammer synchronous vibration system. Hence, electromechanical coupling strengthζ can be defined as(17)
ζ
=
Δ
T
-
C
=
2
λ
θ
¨
1
-
θ
¨
2
ω
0
sin
⁡
2
α
.From (15), it can be seen that the coupling torques of the two motors change in opposite directions and are related to vibration parameters in y and φ directions, phase difference, angular velocity, angular acceleration, and soil parameter. It can be inferred that the vibration system and dynamic system are mutually dependent and affect each other through coupling torque, while the coupling torque in turn affects rotating speed of asynchronous motors.According to (16) and (17), it is easy to find that if the mechanical properties of motor are rigid, namely, motor speed is not easily affected by the coupling torque, electromechanical coupling strength will be small. When phase difference 2
α is ±
π
/
2, electromechanical coupling strength is the largest, and when the phase difference 2
α is zero, the electromechanical coupling strength is zero.In addition, (15) shows that the value and positive of λ directly determine electromechanical coupling characteristics. If the vertical vibration is dominant, λ
≈
B
0
A
y
cos
⁡
γ
y. Furthermore, if excitation frequency is in low frequency region, resonate region, and high frequency region, the domain of phase angle between excitation and response γ
y is 0 to π
/
2, π
/
2, and π
/
2 to π, respectively. Similarly, the influence law of φ direction can be obtained. Moreover, according to working characteristics of motor, the basic electromechanical coupling rules can be obtained through qualitative analysis, as shown in Table 1.Table 1
Basic electromechanical coupling rules.
Excited frequency
Vibrational state
Vertical vibration is dominant (y direction)
Deflection vibrationis dominant (φ direction)
Low frequency region
λ
>
0
⇒
Δ
T
-
L
<
0
⇒
T
-
C
1
↓
,
T
-
C
2
↑
⇒
2
α
→
0
λ
<
0
⇒
Δ
T
-
L
>
0
⇒
T
-
C
1
↑
,
T
-
C
2
↓
⇒
2
α
→
π
Resonate region
λ
→
0
+
⇒
Δ
T
-
L
→
0
-
⇒
T
-
C
1
↓
,
T
-
C
2
↑
⇒
2
α
→
0
+
λ
→
0
-
⇒
Δ
T
-
L
→
0
+
⇒
T
-
C
1
↑
,
T
-
C
2
↓
⇒
2
α
→
π
-
High frequency region
λ
<
0
⇒
Δ
T
-
L
>
0
⇒
T
-
C
1
↑
,
T
-
C
2
↓
⇒
2
α
→
π
λ
>
0
⇒
Δ
T
-
L
<
0
⇒
T
-
C
1
↓
,
T
-
C
2
↑
⇒
2
α
→
0


## 4. Self-Synchronization and Frequency Hopping

Frequency capture and hopping phenomenon for multihammer synchronous vibration system are studied and implemented based on the above sections. According to (15)–(17), due to the electromechanical coupling effect, motors lagging in phase demonstrate drive torque, and motors with phase shifted ahead demonstrate resistance torque. This would be observed, irrespective of the differences in the initial speed of each hammer motor. Moreover, motor speed and phase difference of each hammer tend to be similar gradually, so each vibratory hammer has the same vibration frequency and multihammer synchronous vibration system reaches energy equilibrium. This phenomenon, where vibration frequency of each hammer automatically reaches similar values, is called self-synchronization phenomenon [16, 17], which has been observed in some engineering applications [18].The eccentric block of vibration hammer is driven by the motor. As a result, rotation of eccentric block has a direct mechanical effect on the motor shaft, affecting the operating state of motor. The load analysis diagram of the rotating eccentric block is shown in Figure4(a). Assuming that only vertical vibration exists for the vibration hammer, load torque of motor shaft T
L can be represented as(18)
T
L
=
m
g
-
a
y
r
c
cos
⁡
θ
i
,where θ
i is rotation angle, and θ
i
=
ω
t, a
y is vibration acceleration in vertical direction, and g is acceleration due to gravity.Figure 4
Sketch of frequency hopping phenomenon.
(a)
Force analysis
(b)
Mechanical characteristic curve of three-phase asynchronous motorWhen the multihammer vibration system tends to synchronize, amplitudea
y is increased gradually. Furthermore, according to (18), maximal value of T
L also increases gradually and may exceed the maximum value of electromagnetic torque for a three-phase asynchronous motor. In addition, in this case, sharp decrease in motor speed along with a change in the vibration frequency of this vibratory hammer is expected, as shown in Figure 4(b). This sudden change in frequency due to load torque exceeding the maximum electromagnetic torque (T
m
a
x as shown in Figure 4(b)) of motor is referred to as frequency hopping.The self-synchronization and frequency hopping phenomenon are a complex electromechanical coupling phenomenon, and the essence of the phenomenon is the result of the interaction between the vibration system and the motor system.


## 5. Numerical Simulation of Coupling Process

The simulation model was established according to (3)-(4), and the electromechanical coupling process was obtained by numerical simulation using previously mentioned ideal conditions. During simulation, the initial phase difference was set as π
/
2. Results were obtained when the rotating speeds of the two hammer motors were identical, as shown in Figure 5. From the results, it can be seen that the phase difference gradually converges to zero when motor speed is lower, and the higher the rotation speed, the larger the convergence rate. When the rotation speeds are higher than a certain level relevant to soil parameters, phase difference tends to π, as shown in Figure 5(c). As the phase difference tends towards zero, the twist angle φ gradually attenuates and stabilizes around zero. Also, amplitude in the y direction gradually stabilizes, as shown in Figures 5(a) and 5(b). Figure 5(d) shows simulation results in different initial phase differences; it can be seen that, for a smaller initial phase difference, convergence rate is larger. The simulation results agree well with Table 1.Figure 5
Electromechanical coupling process under the ideal conditions.
(a)
Amplitude aty direction
(b)
Twist angle
(c)
Phase difference at different motor speed
(d)
Phase difference at different initial phase differenceUnder ideal conditions, influence of soil parameters such as equivalent stiffness coefficientsk
y and k
φ along with equivalent damping coefficients c
y and c
φ was simulated, as shown in Figure 6. For this simulation, rotation speeds of the two-hammer motors were set as 1100 r/min. Figure 6(a) shows the phase difference at different values of k
y. It can be observed that, in larger values of k
y, time required for phase difference to tend towards zero was smaller. Figure 6(b) shows the change in the simulated phase difference for different k
φ values. As seen from the figure, for smaller k
φ, convergence rate for phase difference to zero is faster. Figure 6(c) shows the changes in the phase difference for varying c
y. It can be seen that when c
y is smaller, phase difference tends towards a nonzero value, whereas when c
y is larger, phase difference converges to zero. Thus, for the larger values of c
y, a higher convergence rate is expected. However, influence rule of c
φ on self-synchronization is opposite to that of c
y, as shown in Figure 6(d). From the above simulation analysis, it can be seen that soil parameters have large influences on the electromechanical coupling features of the system. The reason is that soil parameters (k
y
,
k
φ
,
c
y
,
c
φ) have a direct influence on vibration state of the vibration system.Figure 6
Simulated phase difference curves under different soil parameters.
(a)
Phase difference under differentk
y
(b)
Phase difference under differentk
φ
(c)
Phase difference under differentc
y
(d)
Phase difference under differentc
φ


## 6. Experiments

In order to verify the correctness and validate conclusion of the above theoretical analysis and simulations, a piling test was carried out. The test equipment is shown in Figure7. Here, two 90 KW electric vibratory hammers were used. Diameter of the pile is 1450 mm, pile length is 9.8 m, wall thickness is 22 mm, and weight is 7700 kg.Figure 7
Photograph of the test equipment.Figure8 shows the recorded phase difference for every 1 m that the pile sinks. It can be seen that the phase difference tends towards a certain value under the action of electromechanical coupling. However, the trend for total change was observed to increase with the sinking of the pile. Figure 9 shows the phase difference plotted when the degree of penetration h of the pile is at 7 m. At this time, the phase difference is about 8
π
/
9, and resultant force of two hammers will be reduced to a very small value, resulting in the pile no longer sinking. The reason of this phenomenon is that deflection and vertical vibration exist in the process of pile sinking, but in the initial stage, deflection of vibration is large relatively, so phase difference tends to zero. With the sinking of pile, electromechanical coupling of vertical vibration increases gradually, so phase difference increases gradually. Obviously, the experimental results agree well with the conclusions in Table 1.Figure 8
Phase difference as a function of pile sinking depth.Figure 9
Phase difference curve ath
=
7 m.Reducing the angleθ for all hammers to zero as shown in Figure 1, eccentric torque of each hammer is the largest. In such a case, rotating speeds of the motors of the two pile hammers are both set as 759 r/min. The self-synchronization and hopping phenomena were observed throughout this experiment, as shown in Figure 10. The phase difference reduced gradually and the two vibration hammers tend to be automatically synchronized within a time period of t
1; this is the self-synchronization phenomenon. When the phase difference reduces to about π
/
3, rotation speeds of the two motors will change in opposite directions, and then the phase difference rapidly increases in the time period of t
2; this is the frequency hopping phenomenon described previously. This phenomenon is observed because as vibration acceleration increases phase difference decreases, resulting in the load torque T
L of the motors increasing along with self-synchronization. When T
L achieves a certain value, motors work in an unstable region, and rotation speed changes suddenly, with phase difference of the two hammers increasing rapidly. Simultaneously, vibration acceleration reduces quickly, T
L is back to normal values, and then the process of self-synchronization is reproduced under the action of electromechanical coupling. This experiment result verifies the existence of self-synchronization and frequency hopping phenomenon intuitively and also proves the correctness of theoretical derivation and simulation results. It provides a theoretical reference for using self-synchronization and avoiding frequency hopping.Figure 10
Self-synchronization and frequency hopping phenomenon.


## 7. Conclusions

Multihammer system is widely used in the pile foundation engineering; the previous method is to connect each hammer using a mechanical structure. Currently, researchers are focused on developing new methods to eliminate the need for these mechanical structures and use control methods to realize the synchronous vibration of multihammer system. This has led to the discovery that complex electromechanical coupling exists in this system, and it is necessary to study electromechanical coupling characteristics of such a system.In this work, we derive and validate the electromechanical coupling properties of a multihammer synchronous vibration system. A mathematical model of multihammer synchronous vibration system was proposed, and electromechanical coupling strength was determined using the small parameter periodic averaging method. Based on this model, basic behavioral rules were obtained; that is, the coupling effect is transmitted through the load torque of the motors and finally automatically reaches torque balance. As a result, phase difference tends to a certain value that is related to the mechanical characteristic curve of three-phase motor and soil parameters. Furthermore, self-synchronization and frequency hopping phenomenon in the multihammer synchronous vibration system are a result of the interaction between the vibration system and the motor system. Self-synchronization is beneficial; however, frequency hopping must be prevented.Both theoretical analysis and experiment demonstrate the correctness of theoretical modeling and analysis. These results provide a theoretical reference for understanding and using of electromechanical coupling of multihammer synchronous vibration system. Furthermore, it also serves as theoretical basis for synchronous control which is designed to eliminate the fixed phase difference caused by electromechanical coupling.



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*Source: 6707264-2016-03-15.xml*