Dynamics of synchronization for four hydraulic motors in a vibrating pile driver system

Abstract

This article proposes a self-synchronous vibrating mechanism with four unbalanced rotors driven by four hydraulic motors mounted symmetrically on two coaxial rotating beams. Analytical expressions of the steady-state response for the multi-body vibrating system are deduced. The dimensionless coupling equations of the four unbalanced rotors are derived by virtue of the averaging method of modified small parameters. The synchronization and stability criteria are deduced using the existence and stability of the zero solution for the dimensionless coupling equations of the four unbalanced rotors. The coefficients of synchronization ability are defined. By numeric analysis, effect of the dynamic parameters on the characteristics of coupling dynamics for the unbalanced rotors is discussed to determine dynamic parameter intervals of synchronization stability. Computer simulations are carried out to verify the correctness of the theoretical analysis.

Introduction

As a kind of construction machinery, the vibratory pile driver is widely used in infrastructure engineering such as urban construction, bridges, and ports. During the working process, the pile hammer drives the pile to make the pile and its surrounding soil in a state of forced vibration. The pile overcomes the resistance and gradually sinks into the soil under the action of the hammer exciting force and the gravity of the hammer and the pile. ¹ The essence of a vibratory pile driver is to reduce the pile friction by vibration. ² At present, most of the engineering vibratory pile drivers use gears to synchronize unbalanced rotors (URs) in the hammer. ³ This rigid synchronous transmission often makes mechanical structure complicated, which needs constant maintenance.

Self-synchronous phenomena of URs, caused by vibration, have opened up fresh opportunities in the vibration utilization technology. This synchronous phenomenon was defined as a property of material objects of very different natures to develop a uniform rhythm of coexistence in spite of differences in individual rhythms. ⁴ Using the method of direct separation of motions and the Poincare–Lyapunov small-parameter method, Blekhman proposed the synchronization theory of vibrating machines with two or more URs and has successfully solved a number of self-synchronization, such as self-synchronization of the unbalanced and planetary URs on a flatly oscillating solid body, self-synchronization of the URs on the spatially oscillating softly vibrato-isolated solid body, self-synchronization of the URs in the carrying vibrating systems with collisions of the bodies, and so on.^4–8 In view of engineering applications, Chinese Scholar Wen applied such methods to develop the theories of self-synchronization of two URs in vibrating systems of circular motion, linear motion, centroid rotation motion, and spatial motion and established a branch of vibration utilization engineering.^9–12

The key to develop vibrating machines with multiple URs lies in the fact that the coupling characteristics of the system and its internal law of the distribution of energy must be understood perfectly. In a single freedom vibrating system with two identical URs rotating in opposite directions, the vibrations excited by the two URs would be canceled mutually and the system was stationary. ⁴ The coupling characteristics of two URs in a rigid base of six freedoms were discussed in detail by the authors in Zhao et al. ¹³ In fact, a vibrating system with two URs has general dynamic symmetry which results in synchronization of the two URs. ¹⁴ Balthazar et al.^15,16 investigated the coupling characteristics of two non-ideal sources (URs) on a flexible portal frame structure and developed four non-ideal sources through a numeric method in which self-synchronization in pre-resonance and resonance regions stemming from the interaction of system responses was also analyzed. Taking three counting-rotating URs, for example, Zhang et al. ¹⁷ discussed numerically the coupling characteristics of multiple URs on a spring-mass rigid base and synchronization regime of the system. Zhao et al. ¹⁸ proposed a new vibrating mechanism of far-resonant vibration driven by four URs, which consisted of a main rigid frame and two accessorial frigid frames and was a far-resonant vibrating system with small damping. The numerical results demonstrated that self-synchronization of the four URs stemmed from the dynamic characteristics of motion selection of the vibrating system.

In accordance with general dynamic symmetry and motion selection principle of the vibrating system, this article proposes a vibratory pile driver driven by four hydraulic motors, which can achieve self-synchronization when the system with a big coefficient of damping operates in a sub-resonance state. The rest of this article is organized as follows: the coupling equations of the four URs are derived in section “Non-dimensional coupling equations of the vibrating pile driver system,” which is followed by the criterion of synchronization and that of stability for its synchronous state in section “Synchronization of the four hydraulic motors.” Numeric discussions are given in section “Numeric analysis and computer simulation” and conclusions are made finally.

Non-dimensional coupling equations of the vibrating pile driver system

Structure of the considered vibrating pile driver

Figure 1 shows the structure of the four-hydraulic motor vibrating pile driver, which consists of pile gripper 1, two beams 2, four hydraulic motors with URs 3, axis 4, upper shell 5, screw nuts 6, and springs 7. There are two hydraulic motors (UR) installed on each beam rotating about axis 4 and there is an angle $\delta$ between the motor axis and axis 4. The two beams are fixed between upper shell 5 and pile gripper 1 by thrust ball bearings. In the working process of a pile machine, the pile driver is installed between a supporting beam and a pile. ¹⁹ Figure 2(a) shows the dynamic model of the vibrating pile system. Figure 2(b) illustrates the distribution of the four URs. Because there are the thrust bearings between $m_1$ and $m_2$ , the body $m_1$ can move in x- and y-directions inside the body $m_2$ and move together with the body $m_2$ in z-direction.

Figure 1.

The structure of exciters in four-hydraulic-motor self-synchronization vibrating pile driver.

Figure 2.

Dynamic model of the self-synchronization vibrating pile driver system: (a) dynamic model of the system and (b) dynamic model of vibration excitors.

Choosing $m_{\mathrm{r}}$ displacement $z_{\mathrm{r}}$ in z-direction, $m_1$ and $m_2$ displacement z in z-direction, $m_1$ displacements x and y in x- and y-directions, the two beams $m_{01}$ and $m_{02}$ swings ${\psi }_1$ , ${\psi }_2$ about z-axis, and the angular displacements, ${\phi }_{11}$ , ${\phi }_{12}$ , ${\phi }_{21}$ , and ${\phi }_{22}$ , of the four URs as the general coordinates and $\mathrm{Q}={\left[\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& T_{\mathrm{m}1}& T_{\mathrm{m}2}& T_{\mathrm{m}3}& T_{\mathrm{m}4}\end{array}\right]}^{\mathrm{T}}$ as the general forces, and using the Lagrange’s equations, the equations of motion of the system are setup as follows

$$\begin{array}{c}(m_1+m_{01}+m_{02}+m_{11}+m_{12}+m_{21}+m_{22})\stackrel{··}{x}+(f_{01x}+f_{02x}+f_x)\stackrel{.}{x}+(k_{01x}+k_{02x}+k_x)x\\ -r\cos \delta (m_{11}\cos {\phi }_{11}+m_{12}\cos {\phi }_{12}){\stackrel{··}{\psi }}_1+2r\cos \delta (m_{11}{\stackrel{.}{\phi }}_{11}\sin {\phi }_{11}+m_{12}{\stackrel{.}{\phi }}_{12}\sin {\phi }_{12}){\stackrel{.}{\psi }}_1\\ +r\cos \delta (m_{11}{\stackrel{.}{\phi }}_{11}^2\cos {\phi }_{11}+m_{11}{\stackrel{··}{\phi }}_{11}\sin {\phi }_{11}+m_{12}{\stackrel{.}{\phi }}_{12}^2\cos {\phi }_{12}+m_{12}{\stackrel{··}{\phi }}_{12}\sin {\phi }_{12}){\psi }_1\\ -(m_{21}r\cos \delta \cos {\phi }_{21}+m_{22}r\cos \delta \cos {\phi }_{22}+m_{21}{l}_{02}-m_{22}{l}_{02}){\stackrel{··}{\psi }}_2\\ +2r\cos \delta (m_{21}{\stackrel{.}{\phi }}_{21}\sin {\phi }_{21}+m_{22}{\stackrel{.}{\phi }}_{22}\sin {\phi }_{22}){\stackrel{.}{\psi }}_2\\ +r\cos \delta (m_{21}{\stackrel{.}{\phi }}_{21}^2\cos {\phi }_{21}+m_{21}{\stackrel{··}{\phi }}_{21}\sin {\phi }_{21}+m_{22}{\stackrel{.}{\phi }}_{22}^2\cos {\phi }_{22}+m_{22}{\stackrel{··}{\phi }}_{22}\sin {\phi }_{22}){\psi }_2\\ =r\cos \delta {\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^2{m}_{ij}({\stackrel{··}{\phi }}_{ij}\cos {\phi }_{ij}-{\stackrel{.}{\phi }}_{ij}^2\sin {\phi }_{ij})}}\\ (m_1+m_{01}+m_{02}+m_{11}+m_{12}+m_{21}+m_{22})\stackrel{··}{y}+(f_{01y}+f_{02y}+f_y)\stackrel{.}{y}+(k_{01y}+k_{02y}+k_y)y\\ -(m_{11}r\sin {\phi }_{11}+m_{12}r\sin {\phi }_{12}-m_{11}{l}_{01}+m_{12}{l}_{01}){\stackrel{··}{\psi }}_1-2r(m_{11}{\stackrel{.}{\phi }}_{11}\cos {\phi }_{11}+m_{12}{\stackrel{.}{\phi }}_{12}\cos {\phi }_{12}){\stackrel{.}{\psi }}_1\\ +r\cos \delta (m_{11}{\stackrel{.}{\phi }}_{11}^2\sin {\phi }_{11}-m_{11}{\stackrel{··}{\phi }}_{11}\cos {\phi }_{11}+m_{12}{\stackrel{.}{\phi }}_{12}^2\sin {\phi }_{12}-m_{12}{\stackrel{··}{\phi }}_{12}\cos {\phi }_{12}){\psi }_1\\ -r\cos \delta (m_{21}\sin {\phi }_{21}+m_{22}\sin {\phi }_{22}){\stackrel{··}{\psi }}_2-2r\cos \delta (m_{21}{\stackrel{.}{\phi }}_{21}\cos {\phi }_{21}+m_{22}{\stackrel{.}{\phi }}_{22}\cos {\phi }_{22}){\stackrel{.}{\psi }}_2\\ +r\cos \delta (m_{21}{\stackrel{.}{\phi }}_{21}^2\sin {\phi }_{21}-m_{21}{\stackrel{··}{\phi }}_{21}\cos {\phi }_{21}+m_{22}{\stackrel{.}{\phi }}_{22}^2\sin {\phi }_{22}-m_{22}{\stackrel{··}{\phi }}_{22}\cos {\phi }_{22}){\psi }_2\\ =r\cos \delta {\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^2{m}_{ij}({\stackrel{··}{\phi }}_{ij}\sin {\phi }_{ij}+{\stackrel{.}{\phi }}_{ij}^2\cos {\phi }_{ij})}}\\ (m_1+m_{01}+m_{02}+m_{11}+m_{12}+m_{21}+m_{22}+m_2)\stackrel{··}{z}+(f_{z1}+f_{z2})\stackrel{.}{z}+(k_{z1}+k_{z2})z\\ -f_{z2}{\stackrel{.}{z}}_r-k_{z2}{z}_r=r\sin \delta {\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^2{(-1)}^i{m}_{ij}({\stackrel{··}{\phi }}_{ij}\sin {\phi }_{ij}+{\stackrel{.}{\phi }}_{ij}^2\cos {\phi }_{ij})}}\\ (m_1+m_{01}+m_{02}+m_{11}+m_{12}+m_{21}+m_{22}+m_2)\stackrel{··}{z}+(f_{z1}+f_{z2})\stackrel{.}{z}+(k_{z1}+k_{z2})z\\ -f_{z2}{\stackrel{.}{z}}_r-k_{z2}{z}_r=r\sin \delta {\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^2{(-1)}^i{m}_{ij}({\stackrel{··}{\phi }}_{ij}\sin {\phi }_{ij}+{\stackrel{.}{\phi }}_2^{ij}\cos {\phi }_{ij})}}\\ m_r{\stackrel{··}{z}}_r+(f_{z2}+f_{z3}){\stackrel{.}{z}}_r+(k_{z2}+k_{z3})z_r-f_{z2}\stackrel{.}{z}-k_{z2}z=0\\ (J_{01}+m_{11}{l}_{01}^2+m_{11}{r}^2+m_{12}{l}_{01}^2+m_{12}{r}^2){\stackrel{··}{\psi }}_1+(f_{01y}{l}_{x1}^2+f_{\mathrm{s}1}){\stackrel{.}{\psi }}_1+k_{01y}{l}_{x1}^2{\psi }_1\\ -r\cos \delta (m_{11}\cos {\phi }_{11}+m_{12}\cos {\phi }_{12})\stackrel{··}{x}-(m_{11}r\sin {\phi }_{11}+m_{12}r\sin {\phi }_{12}-m_{11}{l}_{01}+m_{12}{l}_{01})\stackrel{··}{y}\\ +{\stackrel{··}{\phi }}_{11}{m}_{11}{r}^2\cos \delta +{\stackrel{··}{\phi }}_{12}{m}_{12}{r}^2\cos \delta =r{l}_{01}\cos \delta {\displaystyle \sum _{j=1}^2{(-1)}^{j-1}{m}_{1j}({\stackrel{··}{\phi }}_{1j}\sin {\phi }_{1j}+{\stackrel{.}{\phi }}_{1j}^2\cos {\phi }_{1j})}\\ (J_{02}+m_{21}{l}_{02}^2+m_{21}{r}^2+m_{22}{l}_{02}^2+m_{22}{r}^2){\stackrel{··}{\psi }}_2+(f_{02x}{l}_{y1}^2+f_{\mathrm{s}2}){\stackrel{.}{\psi }}_2+k_{02x}{l}_{y1}^2{\psi }_2\\ -(m_{21}r\cos {\phi }_{21}\cos \delta +m_{22}r\cos {\phi }_{22}\cos \delta +m_{21}{l}_{02}-m_{22}{l}_{02})\stackrel{··}{x}-r\cos \delta (m_{21}\sin {\phi }_{21}+m_{22}\sin {\phi }_{22})\stackrel{··}{y}\\ +{\stackrel{··}{\phi }}_{21}{m}_{21}{r}^2\cos \delta +{\stackrel{··}{\phi }}_{22}{m}_{22}{r}^2\cos \delta =r{l}_{02}{\displaystyle \sum _{j=1}^2{(-1)}^j{m}_{2j}({\stackrel{··}{\phi }}_{2j}\cos {\phi }_{2j}-{\stackrel{.}{\phi }}_{2j}^2\sin {\phi }_{2j})}\\ (J_{11}+m_{11}{r}^2){\stackrel{··}{\phi }}_{11}+f_{11}{\stackrel{.}{\phi }}_{11}=T_{m1}+m_{11}r(\stackrel{··}{x}\cos {\phi }_{11}+\stackrel{··}{y}\sin {\phi }_{11}\cos \delta -\stackrel{··}{z}\sin {\phi }_{11}\sin \delta +{\stackrel{··}{\psi }}_1{l}_{01}\cos \delta \sin {\phi }_{11})\\ (J_{12}+m_{12}{r}^2){\stackrel{··}{\phi }}_{12}+f_{12}{\stackrel{.}{\phi }}_{12}=T_{m2}+m_{12}r(\stackrel{··}{x}\cos {\phi }_{12}+\stackrel{··}{y}\sin {\phi }_{12}\cos \delta -\stackrel{··}{z}\sin {\phi }_{12}\sin \delta -{\stackrel{··}{\psi }}_1{l}_{01}\cos \delta \sin {\phi }_{12})\\ (J_{21}+m_{21}{r}^2){\stackrel{··}{\phi }}_{21}+f_{21}{\stackrel{.}{\phi }}_{21}=T_{m3}+m_{21}r(\stackrel{··}{x}\cos {\phi }_{21}+\stackrel{··}{y}\sin {\phi }_{21}\cos \delta +\stackrel{··}{z}\sin {\phi }_{21}\sin \delta -{\stackrel{··}{\psi }}_2{l}_{02}\cos {\phi }_{21})\\ (J_{22}+m_{22}{r}^2){\stackrel{··}{\phi }}_{22}+f_{22}{\stackrel{.}{\phi }}_{22}=T_{m4}+m_{22}r(\stackrel{··}{x}\cos {\phi }_{22}+\stackrel{··}{y}\sin {\phi }_{22}\cos \delta +\stackrel{··}{z}\sin {\phi }_{22}\sin \delta +{\stackrel{··}{\psi }}_2{l}_{02}\cos {\phi }_{22})\end{array}\}$$

(1)

where $k_{{\psi }_1}=k_{01y}{l}_{x1}^2$ , $k_{\mathrm{xx}}=k_{01x}+k_{02x}+k_x$ , $k_{{\psi }_2}=k_{02x}{l}_{y1}^2$ , $k_z=k_{z1}+k_{z2}$ , $k_{\mathrm{yy}}=k_{01y}+k_{02y}+k_y$ , $k_{z\mathrm{r}}=k_{z2}+k_{z3}$ , $f_{\mathrm{xx}}=f_{01x}+f_{02x}+f_x$ , $f_z=f_{z1}+f_{z2}$ , $f_{\mathrm{yy}}=f_{01y}+f_{02y}+f_y$ , $f_{\mathrm{zr}}=f_{z2}+f_{z3}$ , $f_{{\psi }_1}=f_{01y}{l}_{x1}^2+f_{\mathrm{s}1}$ , and $f_{{\psi }_2}=f_{02x}{l}_{y1}^2+f_{\mathrm{s}2}$ .

When the system operates in a steady state, the phase of the UR ij leads φ by ${\vartheta }_{2(i-1)+j}$ , that is, ${\phi }_{\mathrm{ij}}=\phi +{\vartheta }_{2(i-1)+j}$ , then equation (1) can be rewritten as follows

$$\begin{array}{c}{M}_1\stackrel{··}{x}+f_{xx}\stackrel{.}{x}+k_{xx}x=-m_{0}r{\omega }_{\mathrm{m}0}^2{\displaystyle \sum _{j=1}^4\sin (\phi +{\vartheta }_j)}\\ M_1\stackrel{··}{y}+f_{yy}\stackrel{.}{y}+k_{yy}y=m_{0}r{\omega }_{\mathrm{m}0}^2\cos \delta {\displaystyle \sum _{j=1}^4\cos (\phi +{\vartheta }_j)}\\ M_2\stackrel{··}{z}+f_z\stackrel{.}{z}+k_{z}z-f_{z2}{\stackrel{.}{z}}_r-k_{z2}{z}_r=m_{0}r{\omega }_{\mathrm{m}0}^2\sin \delta {\displaystyle \sum _{j=1}^2[-\cos (\phi +{\vartheta }_j)+c}os(\phi +{\vartheta }_{j+2})]\\ m_{\mathrm{r}}{\stackrel{··}{z}}_{\mathrm{r}}+f_{z\mathrm{r}}{\stackrel{.}{z}}_{\mathrm{r}}+k_{z\mathrm{r}}{z}_{\mathrm{r}}-f_{z2}\stackrel{.}{z}-k_{z2}z=0\\ J_{{\psi }_1}{\stackrel{··}{\psi }}_1+f_{{\psi }_1}{\stackrel{.}{\psi }}_1+k_{{\psi }_1}{\psi }_1=m_{0}r{\omega }_{\mathrm{m}0}^2{l}_{01}\cos \delta {\displaystyle \sum _{j=1}^2{(-1)}^{j-1}\cos (\phi +{\vartheta }_j)}\\ J_{{\psi }_2}{\stackrel{··}{\psi }}_2+f_{{\psi }_2}{\stackrel{.}{\psi }}_2+k_{{\psi }_2}{\psi }_2=m_{0}r{\omega }_{\mathrm{m}0}^2{l}_{02}{\displaystyle \sum _{j=3}^4{(-1)}^{j-1}\sin (\phi +{\vartheta }_j)}\\ (J_{11}+m_0{r}^2){\stackrel{··}{\phi }}_{11}+f_{11}{\stackrel{.}{\phi }}_{11}=T_{m1}+m_{0}r(\stackrel{··}{x}\cos {\phi }_{11}+\stackrel{··}{y}\sin {\phi }_{11}\cos \delta -\stackrel{··}{z}\sin {\phi }_{11}\sin \delta +{\stackrel{··}{\psi }}_1{l}_{01}\cos \delta \sin {\phi }_{11})\\ (J_{12}+m_0{r}^2){\stackrel{··}{\phi }}_{12}+f_{12}{\stackrel{.}{\phi }}_{12}=T_{m2}+m_{0}r(\stackrel{··}{x}\cos {\phi }_{12}+\stackrel{··}{y}\sin {\phi }_{12}\cos \delta -\stackrel{··}{z}\sin {\phi }_{12}\sin \delta -{\stackrel{··}{\psi }}_1{l}_{01}\cos \delta \sin {\phi }_{12})\\ (J_{21}+m_0{r}^2){\stackrel{··}{\phi }}_{21}+f_{21}{\stackrel{.}{\phi }}_{21}=T_{m3}+m_{0}r(\stackrel{··}{x}\cos {\phi }_{21}+\stackrel{··}{y}\sin {\phi }_{21}\cos \delta +\stackrel{··}{z}\sin {\phi }_{21}\sin \delta -{\stackrel{··}{\psi }}_2{l}_{02}\cos {\phi }_{21})\\ (J_{22}+m_0{r}^2){\stackrel{··}{\phi }}_{22}+f_{22}{\stackrel{.}{\phi }}_{22}=T_{m4}+m_{0}r(\stackrel{··}{x}\cos {\phi }_{22}+\stackrel{··}{y}\sin {\phi }_{22}\cos \delta +\stackrel{··}{z}\sin {\phi }_{22}\sin \delta +{\stackrel{··}{\psi }}_2{l}_{02}\cos {\phi }_{22})\end{array}\}$$

(2)

Torque of a hydraulic motor

Figure 3 shows the schematic diagram of hydraulic system for the vibrating pile driver, which includes four hydraulic circuits. Each circuit consists of an asynchronous motor, a gear bump, a magnetic exchange valve, and a hydraulic motor. The angular velocity relation between the asynchronous motor and the hydraulic motor can be expressed as follows ²⁰

$${\omega }_{\mathrm{e}}={\omega }_{\mathrm{b}}=\frac{V_m}{{\eta }_{\mathrm{mv}}{\eta }_{\mathrm{bv}}{V}_{\mathrm{b}}}{\omega }_{\mathrm{m}}$$

(3)

Figure 3.

Hydraulic system of the vibrating pile driver system: 1-asynchronous motor; 2-hydraulic bump; 3-overflow valve; 4-filter; 5-one-way valve; 6-Magnetic Exchange Valve; 7-hydraulic motor and 8-cooler.

According to Zhao et al., ¹³ the electromagnetic torque of an asynchronous motor operating steadily at the angular velocity, ${\omega }_{\mathrm{m}0}$ , can be expressed as follows

$$T_{\mathrm{e}}=T_{\mathrm{e}0}-k_{\mathrm{e}0}\nu$$

(4)

The driving torque of the bump and the output torque of the hydraulic motor are expressed as follows

$$T_{\mathrm{b}}=\frac{p_{\mathrm{p}}{V}_{\mathrm{b}}}{2\pi {\eta }_{\mathrm{bm}}}$$

(5)

$$T_{\mathrm{m}0}=\frac{p_{\mathrm{m}}{V}_{\mathrm{m}}{\eta }_{\mathrm{bm}}{\eta }_{\mathrm{mm}}}{2\pi }$$

(6)

Because $T_{\mathrm{e}}=T_{\mathrm{b}}$ , inserting equation (4) into equation (5) yields

$$p_{\mathrm{p}}=\frac{2\pi {\eta }_{\mathrm{bm}}(T_{\mathrm{e}0}-k_{\mathrm{e}0}\nu )}{V_{\mathrm{b}}}$$

(7)

Considering equation (3) and inserting equation (7) into equation (6) yield

$$T_{\mathrm{m}}=T_{\mathrm{m}0}-k_{\mathrm{m}0}{\nu }_{\mathrm{m}}$$

(8)

Non-dimensional coupling equation of the four exciters

It is assumed that $\stackrel{·}{\phi }={\omega }_{\mathrm{m}}$ and ${\stackrel{·}{\vartheta }}_i={\nu }_{\mathrm{m}i}{\omega }_{\mathrm{m}}$ , then ${\nu }_{\mathrm{m}i}<<1$ when the system operates at the steady state. The effects of disturbance coefficients ${\nu }_{\mathrm{m}i}$ on the responses of the system can be neglected. ¹³ Then, steady-state responses of the system can be approximately expressed as follows

$$\begin{array}{c}x=-r{r}_{\mathrm{m}}{\mu }_x\sum _{i=1}^4\sin (\phi +{\vartheta }_i-{\gamma }_x)\hfill \\ y=r{r}_{\mathrm{m}}{\mu }_y\cos \delta \sum _{i=1}^4\cos (\phi +{\vartheta }_i-{\gamma }_y)\hfill \\ z=r{r}_{\mathrm{m}}{r}_{\mathrm{m}z}{\mu }_z\sin \delta \sum _{j=1}^2[-\cos (\phi +{\vartheta }_j-{\gamma }_z)+\cos (\phi +{\vartheta }_{j+2}-{\gamma }_z)]\hfill \\ z_{\mathrm{r}}=r{r}_{\mathrm{m}}{r}_{\mathrm{m}z}{\mu }_{z\mathrm{r}}\sin \delta \sum _{j=1}^2[-\cos (\phi +{\vartheta }_j-{\gamma }_{z\mathrm{r}})+\cos (\phi +{\vartheta }_{j+2}-{\gamma }_{z\mathrm{r}})]\hfill \\ {\psi }_1=\frac{r{r}_{\mathrm{m}}{r}_{\mathrm{m}1}{r}_{l1}}{l_{\mathrm{e}1}}{\mu }_{\psi 1}\cos \delta \sum _{j=1}^2{(-1)}^{j-1}\cos (\phi +{\vartheta }_j-{\gamma }_{\psi 1})\hfill \\ {\psi }_2=\frac{r{r}_{\mathrm{m}}{r}_{\mathrm{m}2}{r}_{l2}}{l_{\mathrm{e}2}}{\mu }_{\psi 2}\sum _{j=3}^4{(-1)}^{j-1}\sin (\phi +{\vartheta }_j-{\gamma }_{\psi 2})\hfill \end{array}\}$$

(9)

The symbols in equation (9) are listed in Appendix 2.

Assuming

$$\begin{array}{c}2{\alpha }_1=\frac{1}{2}(\sum _{j=1}^2{\phi }_{1j}-\sum _{j=1}^2{\phi }_{2j})\hfill \\ \multicolumn{1}{c}{2{\alpha }_2=({\phi }_{11}-{\phi }_{12})}\\ \multicolumn{1}{c}{2{\alpha }_3=({\phi }_{21}-{\phi }_{22})}\end{array}\}$$

(10)

then we have

$$\begin{array}{c}{\phi }_{11}=\phi +{\vartheta }_1=\phi +{\alpha }_1+{\alpha }_2\hfill \\ \multicolumn{1}{c}{{\phi }_{12}=\phi +{\vartheta }_2=\phi +{\alpha }_1-{\alpha }_2}\\ \multicolumn{1}{c}{{\phi }_{21}=\phi +{\vartheta }_3=\phi -{\alpha }_1+{\alpha }_3}\\ \multicolumn{1}{c}{{\phi }_{22}=\phi +{\vartheta }_4=\phi -{\alpha }_1-{\alpha }_3}\\ \multicolumn{1}{c}{{\vartheta }_1-{\vartheta }_2=2{\alpha }_2}\\ \multicolumn{1}{c}{{\vartheta }_1-{\vartheta }_3=2{\alpha }_1+{\alpha }_2-{\alpha }_3}\\ \multicolumn{1}{c}{{\vartheta }_1-{\vartheta }_4=2{\alpha }_1+{\alpha }_2+{\alpha }_3}\\ \multicolumn{1}{c}{{\vartheta }_2-{\vartheta }_3=2{\alpha }_1-{\alpha }_2-{\alpha }_3}\\ \multicolumn{1}{c}{{\vartheta }_2-{\vartheta }_4=2{\alpha }_1-{\alpha }_2+{\alpha }_3}\\ \multicolumn{1}{c}{{\vartheta }_3-{\vartheta }_4=2{\alpha }_3}\end{array}\}$$

(11)

Differentiating x, y, z and ${\psi }_1$ , ${\psi }_2$ in equation (9) with respect to time t by the chain rule (noticing ${\stackrel{·}{\phi }}_{\mathrm{ik}}=(1+v_j){\omega }_{\mathrm{m}},i=1,2,j=1,2,3,4,k=1,2$ ), we obtain the second derivative of x, y, z and ${\psi }_1$ , ${\psi }_2$ . Substituting them into the differential equations of motion of the URs in equation (1) and integrating them with $0-2\pi$ to obtain their average equations over $0-2\pi$ , respectively, neglecting the terms of higher orders of ${\overline{v}}_j$ , ${\stackrel{·}{\overline{v}}}_j$ , we obtain

$$(J_{ij}+m_0{r}^2){\omega }_{\mathrm{m}}{\stackrel{.}{\overline{\nu }}}_l+f_{ij}{\omega }_{\mathrm{m}}(1+{\overline{\nu }}_l)=T_{\mathrm{m}l}-T_{\mathrm{L}l},l=2(i-1)+j;i,j=1,2$$

(12)

The symbols in equation (12) are listed in Appendix 3. For each hydraulic motor, its output torque in equation (8) can be expressed as follows

$$T_{\mathrm{m}i}=T_{\mathrm{m}0i}-k_{\mathrm{m}0i}{\nu }_{\mathrm{m}i},i=1,2,3,4$$

(13)

The moments of inertia of the motors are much less than the moments of inertia of the URs. Hence, $J_{\mathrm{ij}}$ can be neglected. Here, assuming $f_1=f_{11}$ , $f_2=f_{12}$ , $f_3=f_{21}$ , $f_4=f_{22}$ , then dividing both sides of equations (12) by $m_0{r}^2{\omega }_{\mathrm{m}0}$ and writing them into the matrix form, we have

$$\mathit{A}\stackrel{·}{\overline{\mathit{v}}}=\mathit{B}\overline{\mathit{v}}+\mathit{u}$$

(14)

with

$$\begin{array}{c}\overline{\mathit{v}}={[\begin{array}{cccc}{\overline{v}}_1\hfill & {\overline{v}}_2\hfill & {\overline{v}}_3\hfill & {\overline{v}}_4\hfill \end{array}]}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{\mathit{u}={[\begin{array}{cccc}{u}_1\hfill & u_2\hfill & u_3\hfill & u_4\hfill \end{array}]}^{\mathrm{T}}}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathit{A}=[a_{\mathrm{ij}}{]}_{4\times 4}}\\ \multicolumn{1}{c}{\mathit{B}=[b_{\mathrm{ij}}{]}_{4\times 4}}\end{array}$$

$$a_{\mathrm{ij}}=\{\begin{array}{c}1+\frac{1}{2}{W}_{\mathrm{c}\mathrm{ii}},i=j\hfill \\ \multicolumn{1}{c}{\frac{1}{2}[W_{\mathrm{c}\mathrm{ij}}\cos ({\overline{\vartheta }}_i-{\overline{\vartheta }}_j)-W_{\mathrm{s}\mathrm{ij}}\sin ({\overline{\vartheta }}_i-{\overline{\vartheta }}_j)],i\ne j}\end{array}$$

$$b_{\mathrm{ij}}=\{\begin{array}{c}-\frac{f_i}{m_0{r}^2}-{\omega }_{\mathrm{m}0}{W}_{\mathrm{s}\mathrm{ii}}-\frac{k_{\mathrm{m}0i}}{m_0{r}^2{\omega }_{\mathrm{m}0}},i=j\hfill \\ \multicolumn{1}{c}{-{\omega }_{\mathrm{m}0}[W_{\mathrm{s}\mathrm{ij}}\cos ({\overline{\vartheta }}_i-{\overline{\vartheta }}_j))+W_{\mathrm{c}\mathrm{ij}}\sin ({\overline{\vartheta }}_i-{\overline{\vartheta }}_j))],i\ne j}\end{array}$$

$$u_i=\frac{T_{\mathrm{m}0i}-f_i{\omega }_{\mathrm{m}0}}{m_0{r}^2{\omega }_{\mathrm{m}0}}-\frac{1}{2}{\omega }_{\mathrm{m}0}{\displaystyle \sum _{\mathrm{j}=1}^4[W_{\mathrm{s}ij}\cos ({\overline{\vartheta }}_i-{\overline{\vartheta }}_j)+W_{\mathrm{c}ij}\sin ({\overline{\vartheta }}_i-{\overline{\vartheta }}_j)]},i=1,2,3,4$$

Equation (14) describes the coupling relation of four URs and is called the non-dimensional coupling equation of the four URs, where matrix A is relative to the moments of inertia of four exciters, which can be referred to as the non-dimensional inertia coupling matrix of four exciters. Matrix B is relative to the stiffness of angular velocity of four motors and referred to as the non-dimensional stiffness coupling matrix of four motors. $\overline{\mathit{v}}$ is the non-dimensional disturbance parameters of angular velocities of the URs around ${\omega }_{\mathrm{m}0}$ . Vector u represents the non-dimensional load torques of the four hydraulic motors.

Synchronization of the four hydraulic motors

Synchronization criterion

When the system reaches the synchronization, the average coefficient of angular velocity instantaneous fluctuation in a single period of the URs is 0. Assuming that $\overline{\nu }=0$ , then we obtain u  = 0 from equation (14), that is

$$T_{\mathrm{o}1}=T_{\mathrm{u}}\sum _{j=1}^4[W_{\mathrm{s}1j}\cos ({\overline{\vartheta }}_1-{\overline{\vartheta }}_j)+W_{\mathrm{c}1j}\sin ({\overline{\vartheta }}_1-{\overline{\vartheta }}_j)]$$

(15)

$$T_{\mathrm{o}2}=T_{\mathrm{u}}\sum _{j=1}^4[W_{\mathrm{s}2j}\cos ({\overline{\vartheta }}_2-{\overline{\vartheta }}_j)+W_{\mathrm{c}2j}\sin ({\overline{\vartheta }}_2-{\overline{\vartheta }}_j)]$$

(16)

$$T_{\mathrm{o}3}=T_{\mathrm{u}}\sum _{j=1}^4[W_{\mathrm{s}3j}\cos ({\overline{\vartheta }}_3-{\overline{\vartheta }}_j)+W_{\mathrm{c}3j}\sin ({\overline{\vartheta }}_3-{\overline{\vartheta }}_j)]$$

(17)

$$T_{\mathrm{o}4}=T_{\mathrm{u}}\sum _{j=1}^4[W_{\mathrm{s}4j}\cos ({\overline{\vartheta }}_4-{\overline{\vartheta }}_j)+W_{\mathrm{c}4j}\sin ({\overline{\vartheta }}_4-{\overline{\vartheta }}_j)]$$

(18)

Subtracting equation (15) from equation (16), we obtain

$$\Delta T_{\mathrm{o}1}=2T_{\mathrm{u}}(W_{\mathrm{c}12}\sin 2{\overline{\alpha }}_2-2W_{\mathrm{s}13}\sin 2{\overline{\alpha }}_1\sin {\overline{\alpha }}_{2}cos{\overline{\alpha }}_3+2W_{\mathrm{c}13}\cos 2{\overline{\alpha }}_1\sin {\overline{\alpha }}_{2}cos{\overline{\alpha }}_3)$$

(19)

Subtracting equation (17) from equation (18), we obtain

$$\Delta T_{\mathrm{o}2}=2T_{\mathrm{u}}(W_{\mathrm{c}34}\sin 2{\overline{\alpha }}_3+2W_{\mathrm{s}13}\sin 2{\overline{\alpha }}_1\cos {\overline{\alpha }}_2\sin {\overline{\alpha }}_3+2W_{\mathrm{c}13}\cos 2{\overline{\alpha }}_1\cos {\overline{\alpha }}_2\sin {\overline{\alpha }}_3)$$

(20)

Subtracting the sum of equations (17) and (18) from that of equations (15) and (16), we obtain

$$\Delta T_{\mathrm{o}}=2T_{\mathrm{u}}(W_{\mathrm{s}11}-W_{\mathrm{s}33}+W_{\mathrm{s}12}cos2{\overline{\alpha }}_2-W_{\mathrm{s}34}cos2{\overline{\alpha }}_3+4W_{\mathrm{c}13}\sin 2{\overline{\alpha }}_1\cos {\overline{\alpha }}_2\cos {\overline{\alpha }}_3)$$

(21)

Summing up equations (15)–(18), we obtain

$$T_{\mathrm{L}}=2T_{\mathrm{u}}(W_{\mathrm{s}11}+W_{\mathrm{s}33}+W_{\mathrm{s}12}\cos 2{\overline{\alpha }}_2+W_{\mathrm{s}34}\cos 2{\overline{\alpha }}_3+4W_{\mathrm{s}13}\cos 2{\overline{\alpha }}_1\cos {\overline{\alpha }}_2\cos {\overline{\alpha }}_3)$$

(22)

If the parameters of the vibrating system can satisfy a certain condition, equations (19)–(22) can be solved for ${\overline{\alpha }}_1$ , ${\overline{\alpha }}_2$ , ${\overline{\alpha }}_3$ , and ${\omega }_{\mathrm{m}}$ by the numeric method, which are denoted by ${\overline{\alpha }}_{10}$ , ${\overline{\alpha }}_{20}$ , ${\overline{\alpha }}_{30}$ , and ${\omega }_{\mathrm{m}0}$ , respectively. Compared with $W_{\mathrm{c}12}$ , $W_{\mathrm{c}13}$ , and $W_{\mathrm{c}34}$ , $W_{\mathrm{s}11}$ , $W_{\mathrm{s}33}$ , $W_{\mathrm{s}12}$ , $W_{\mathrm{s}13}$ , and $W_{\mathrm{s}34}$ can be neglected in equations (19)–(21). Then, equations (19)–(21) can be simplified as follows

$$\begin{array}{c}({\eta }_1\cos {\overline{\alpha }}_{20}+\cos 2{\overline{\alpha }}_{10}\cos {\overline{\alpha }}_{30})\sin {\overline{\alpha }}_{20}=D_1,\hfill \\ \multicolumn{1}{c}{({\eta }_2\cos {\overline{\alpha }}_{30}+\cos 2{\overline{\alpha }}_{10}\cos {\overline{\alpha }}_{20})\sin {\overline{\alpha }}_{30}=D_2,}\\ \multicolumn{1}{c}{\sin 2{\overline{\alpha }}_{10}\cos {\overline{\alpha }}_{20}\cos {\overline{\alpha }}_{30}=D_0}\end{array}\}$$

(23)

where ${\eta }_1=(W_{\mathrm{c}12}/W_{\mathrm{c}13})$ , ${\eta }_2=(W_{\mathrm{c}34}/W_{\mathrm{c}13})$ , $D_1=(\Delta T_{\mathrm{o}1}/(4T_{\mathrm{u}}{W}_{\mathrm{c}13}))$ , $D_2=(\Delta T_{\mathrm{o}2}/(4T_{\mathrm{u}}{W}_{\mathrm{c}13}))$ , $D_0=(\Delta T_{\mathrm{o}}/(8T_{\mathrm{u}}{W}_{\mathrm{c}13}))$ .

The conditions of implementing the synchronization of the four URs can be expressed as follows

$$\begin{array}{c}\left|D_0\right|<1\hfill \\ \multicolumn{1}{c}{D_1>\frac{(\sqrt{8{\eta }_1^2+1}+3)\sqrt{8{\eta }_1^2+2\sqrt{8{\eta }_1^2+1}-2}}{16\left|{\eta }_1\right|}}\\ \multicolumn{1}{c}{D_2>\frac{(\sqrt{8{\eta }_2^2+1}+3)\sqrt{8{\eta }_2^2+2\sqrt{8{\eta }_2^2+1}-2}}{16\left|{\eta }_2\right|}}\end{array}\}$$

(24)

Stability of synchronization

When u  = 0, equation (14) is the generalized system

$$\mathit{A}\prime \stackrel{·}{\mathit{v}}=\mathit{B}\prime \mathit{v}$$

(25)

where $\mathit{A}\prime$ and $\mathit{B}\prime$ denote the values of A and B for ${\overline{\alpha }}_1={\overline{\alpha }}_{10}$ , ${\overline{\alpha }}_2={\overline{\alpha }}_{20}$ , ${\overline{\alpha }}_3={\overline{\alpha }}_{30}$ , and ${\omega }_{\mathrm{m}}={\omega }_{\mathrm{m}0}$ .

Obviously, $det(\mathit{A}\prime -\mathit{B}\prime )\ne 0$ , and the generalized system described by equation (24) is regular. ¹⁴ $W_{\mathrm{s}\mathrm{ij}}$ is so small that can be neglected, then matrix A is a symmetric matrix, matrix B is an anti-symmetric matrix. Hence, when the system satisfies the conditions equation (24), matrices $\mathit{A}\prime$ and $\mathit{B}\prime$ in equation (25) satisfy the Lyapunov stability criterion^13,14

$$\begin{array}{c}det({\mathit{A}\prime }_2)>0,det({\mathit{A}\prime }_3)>0,\hfill \\ \multicolumn{1}{c}{det(\mathit{A}\prime )>0,}\\ \multicolumn{1}{c}{{a\prime }_{\mathrm{ij}}>0,i=1,2,3,4;j=1,2,3,4}\end{array}\}$$

(26)

$$\begin{array}{c}{\mathit{I}}^{\mathrm{T}}\mathit{B}\prime +{\mathit{B}\prime }^{\mathrm{T}}\mathit{I}=-2\mathrm{diag}\{-b_{11},-b_{22},-b_{33},-b_{44}\},\hfill \\ \multicolumn{1}{c}{{\mathit{A}\prime }^T\mathit{I}=\mathit{I}\mathit{A}\prime >0}\end{array}\}$$

(27)

If $\underset{t\to \infty }{lim}\overline{\mathit{v}}=0$ , the system described by equation (25) is stable. According to equation (11), the stability of $\overline{\mathit{v}}$ is same to the stability of ${\overline{\epsilon }}_i(i=0,1,2,3)$ .

Linearizing equation (12) around ${\overline{\alpha }}_{10}$ , ${\overline{\alpha }}_{20}$ , ${\overline{\alpha }}_{30}$ and neglecting $W_{\mathrm{s}\mathrm{ij}}$ , $f_1$ , $f_2$ , $f_3$ , $f_4$ , and high-order items that higher than the second-order items, then we obtain

$$k_{\mathrm{m}01}({\overline{\epsilon }}_0+{\overline{\epsilon }}_1+{\overline{\epsilon }}_2)=-\sum _{i=1}^3{\left(\frac{\partial {\chi }_{\mathrm{a}1}}{\partial {\overline{\alpha }}_i}\right)}^{*}\Delta {\overline{\alpha }}_i$$

(28)

$$k_{\mathrm{m}02}({\overline{\epsilon }}_0+{\overline{\epsilon }}_1-{\overline{\epsilon }}_2)=-\sum _{i=1}^3{\left(\frac{\partial {\chi }_{\mathrm{a}2}}{\partial {\overline{\alpha }}_i}\right)}^{*}\Delta {\overline{\alpha }}_i$$

(29)

$$k_{\mathrm{m}03}({\overline{\epsilon }}_0-{\overline{\epsilon }}_1+{\overline{\epsilon }}_3)=-\sum _{i=1}^3{\left(\frac{\partial {\chi }_{\mathrm{a}3}}{\partial {\overline{\alpha }}_i}\right)}^{*}\Delta {\overline{\alpha }}_i$$

(30)

$$k_{\mathrm{m}04}({\overline{\epsilon }}_0-{\overline{\epsilon }}_1-{\overline{\epsilon }}_3)=-\sum _{i=1}^3{\left(\frac{\partial {\chi }_{\mathrm{a}4}}{\partial {\overline{\alpha }}_i}\right)}^{*}\Delta {\overline{\alpha }}_i$$

(31)

where $\Delta {\overline{\alpha }}_i={\overline{\alpha }}_i-{\overline{\alpha }}_i^{*},i=1,2,3$ , and $(\partial {\chi }_{\mathrm{a}j}/\partial {\overline{\alpha }}_i{)}^{*}$ denotes the values for ${\omega }_{\mathrm{m}0}^{*}$ , ${\overline{\alpha }}_{10}$ , ${\overline{\alpha }}_{20}$ , ${\overline{\alpha }}_{30}$ , and ${\chi }_{\mathrm{a}i}=T_{\mathrm{u}}\sum _{j=1}^4[W_{\mathrm{c}\mathrm{ij}}\sin ({\overline{\vartheta }}_i-{\overline{\vartheta }}_j)],i=1,2,3,4$ .

We can obtain the following equations through the transition of equations (27)–(30)

$${\overline{\epsilon }}_0=-\frac{1}{4}\sum _{j=1}^4\left(\frac{1}{k_{\mathrm{m}0j}}{\sum _{i=1}^3\left(\frac{\partial {\chi }_{\mathrm{a}j}}{\partial {\overline{\alpha }}_i}\right)}^{*}\Delta {\overline{\alpha }}_i\right)$$

(32)

$${\overline{\epsilon }}_1=-\frac{1}{4}{\displaystyle \sum _{i=1}^3\left(\frac{1}{k_{\mathrm{m}01}}{\left(\frac{\partial {\chi }_{\mathrm{a}1}}{\partial {\overline{\alpha }}_i}\right)}^{\ast }+\frac{1}{k_{\mathrm{m}02}}{\left(\frac{\partial {\chi }_{\mathrm{a}2}}{\partial {\overline{\alpha }}_i}\right)}^{\ast }-\frac{1}{k_{\mathrm{m}03}}{\left(\frac{\partial {\chi }_{\mathrm{a}3}}{\partial {\overline{\alpha }}_i}\right)}^{\ast }-\frac{1}{k_{\mathrm{m}04}}{\left(\frac{\partial {\chi }_{\mathrm{a}4}}{\partial {\overline{\alpha }}_i}\right)}^{\ast }\right)}\Delta {\overline{\alpha }}_i$$

(33)

$${\overline{\epsilon }}_2=-\frac{1}{2}\sum _{i=1}^3(\frac{1}{k_{\mathrm{m}01}}{\left(\frac{\partial {\chi }_{\mathrm{a}1}}{\partial {\overline{\alpha }}_i}\right)}^{*}-\frac{1}{k_{\mathrm{m}02}}{\left(\frac{\partial {\chi }_{\mathrm{a}2}}{\partial {\overline{\alpha }}_i}\right)}^{*})\Delta {\overline{\alpha }}_i$$

(34)

$${\overline{\epsilon }}_3=-\frac{1}{2}\sum _{i=1}^3(\frac{1}{k_{\mathrm{m}03}}{\left(\frac{\partial {\chi }_{\mathrm{a}3}}{\partial {\overline{\alpha }}_i}\right)}^{*}-\frac{1}{k_{\mathrm{m}04}}{\left(\frac{\partial {\chi }_{\mathrm{a}4}}{\partial {\overline{\alpha }}_i}\right)}^{*})\Delta {\overline{\alpha }}_i$$

(35)

Taking $\Delta {\stackrel{·}{\overline{\alpha }}}_i={\stackrel{·}{\overline{\alpha }}}_i={\overline{\epsilon }}_i{\omega }_{\mathrm{m}0}^{*},i=1,2,3$ into account, we can find that equations (33)–(35) are the differential equations of $\Delta \overline{\mathbf{\alpha }}=(\Delta {\overline{\alpha }}_1\Delta {\overline{\alpha }}_2\Delta {\overline{\alpha }}_3{)}^{\mathrm{T}}$ . The matrix form of the three equations is as follows

$$\Delta \stackrel{·}{\overline{\mathbf{\alpha }}}=\mathit{C}\Delta \overline{\mathbf{\alpha }}$$

(36)

According to $\det (\mathit{C}-\lambda \mathit{I})=0$ , the characteristic equation of matrix C can be obtained as follows

$${\lambda }^3+d_1{\lambda }^2+d_2\lambda +d_3=0$$

(37)

The zero solutions of equation (36) are stable only if all the roots in equation (37) have the negative real parts. According to the Routh–Hurwitz stability criterion, we can find that the parameters in equation (37) should satisfy

$$d_1>0,d_3>0,d_1{d}_2>d_3$$

(38)

with $d_1=-c_{11}-c_{22}-c_{33}$ , $d_2=-c_{12}{c}_{21}-c_{23}{c}_{32}-c_{13}{c}_{31}+c_{11}{c}_{22}+c_{22}{c}_{33}+c_{33}{c}_{11}$ , $d_3=-c_{11}{c}_{22}{c}_{33}-c_{12}{c}_{23}{c}_{31}-c_{13}{c}_{21}{c}_{32}+c_{11}{c}_{23}{c}_{32}+c_{22}{c}_{13}{c}_{31}+c_{33}{c}_{12}{c}_{21}$ .

If $\underset{t\to \infty }{lim}\Delta \overline{\alpha }=0$ , also mean $\underset{t\to \infty }{lim}{\overline{\epsilon }}_i=0,i=0,1,2,3$ , according to equation (11), we can have $\underset{t\to \infty }{lim}\overline{\mathit{v}}=0$ . The system implements the synchronization if it satisfies both the Lyapunov stability criterion and the Routh–Hurwitz stability criterion.

In engineering, the parameters of the four hydraulic circuits in Figure 3 are usually chosen to be similar. Hence, equation (26) can be expressed as follows

$$\begin{array}{c}{H}_1=det({\mathit{A}\prime }_1)=1+\frac{1}{2}{W}_{\mathrm{c}11}>0,\hfill \\ \multicolumn{1}{c}{H_2=det({\mathit{A}\prime }_2)=(1+\frac{1}{2}{W}_{\mathrm{c}11})(1+\frac{1}{2}{W}_{\mathrm{c}12})-{\left(\frac{1}{2}{W}_{\mathrm{c}12}\right)}^2>0,}\\ \multicolumn{1}{c}{H_3=det({\mathit{A}\prime }_3)>0,}\\ \multicolumn{1}{c}{H_4=det(\mathit{A}\prime )>0,}\\ \multicolumn{1}{c}{W_{\mathrm{c}12}>0,W_{\mathrm{c}13}\cos 2{{\overline{\alpha }}_1}^{*}>0}\end{array}\}$$

(39)

We can prove that if the system satisfies the Lyapunov stability criterion equation (38), then the system also satisfies the Routh–Hurwitz stability criterion. Therefore, when the parameters of the hydraulic circuits in Figure 3 are similar, the stability criterion of synchronization is equation (39).

Numeric analysis and computer simulation

Ability of synchronization and stability

In order to expediently describe the ability of synchronization of the mechanism proposed in this article, we define the three coefficients of synchronous ability as follows ¹⁸

$${\zeta }_1=\{\begin{array}{c}\multicolumn{1}{c}{\frac{-W_{\mathrm{c}12}}{W_{\mathrm{s}11}-W_{\mathrm{s}12}},W_{\mathrm{c}12}<0}\\ \multicolumn{1}{c}{\frac{W_{\mathrm{c}12}}{W_{\mathrm{s}11}+W_{\mathrm{s}12}},W_{\mathrm{c}12}>0}\end{array}$$

(40)

$${\zeta }_2=\{\begin{array}{c}\multicolumn{1}{c}{\frac{-W_{\mathrm{c}34}}{W_{\mathrm{s}33}-W_{\mathrm{s}34}},W_{\mathrm{c}34}<0}\\ \multicolumn{1}{c}{\frac{W_{\mathrm{c}34}}{W_{\mathrm{s}33}+W_{\mathrm{s}34}},W_{\mathrm{c}34}>0}\end{array}$$

(41)

$$\zeta =\{\begin{array}{c}\multicolumn{1}{c}{\frac{-4W_{\mathrm{c}13}}{W_{\mathrm{s}11}+W_{\mathrm{s}33}+W_{\mathrm{s}12}+W_{\mathrm{s}34}-4W_{\mathrm{s}13}},W_{\mathrm{c}13}<0}\\ \multicolumn{1}{c}{\frac{4W_{\mathrm{c}13}}{W_{\mathrm{s}11}+W_{\mathrm{s}33}+W_{\mathrm{s}12}+W_{\mathrm{s}34}+4W_{\mathrm{s}13}},W_{\mathrm{c}13}>0}\end{array}$$

(42)

When the system is working, the damping of the pile and the soil, z-freedom, is larger, while the dumping of the other freedoms is surly the type of spring-mass. So we assume that ${\xi }_z=0.7$ ² and ${\xi }_x={\xi }_y={\xi }_{z\mathrm{r}}={\xi }_{\psi 1}={\xi }_{\psi 2}=0.07$ . ⁹ The aim of the design for a vibrating pile drive is that the system vibrates in z-freedom, and the motions for the other freedoms are not required. Therefore, we assume that $n_{z2}=6.0$ and $m_{\mathrm{r}}=0.5m_2$ . According to the dynamic characteristics of motion selection of the vibrating system, the two rotating joints, ψ₁- and ψ₂-freedoms, and the two translational freedoms, x- and y-freedoms, are designed to be far-resonance, that is, $n_{\psi 1}=n_{\psi 2}=4.0$ and $n_x=n_y=4.0$ , then we have ${\gamma }_{\psi 1}\in (\pi /2,\pi )$ , ${\gamma }_{\psi 2}\in (\pi /2,\pi )$ , and ${\gamma }_x\in (\pi /2,\pi )$ , ${\gamma }_y\in (\pi /2,\pi )$ . For the stiffness of soil is small, then $n_z<1$ . According to the expressions of $W_{\mathrm{c}12}$ , $W_{\mathrm{c}34}$ , and $W_{\mathrm{c}13}$ , the phase difference for the single pair of the URs on beams 1 or 2 is driven close to zero, and the phase difference between the two pairs of the URs is driven close to $\pi$ . ¹⁸ The other parameters are as follows: $n_z=0.9$ , $r_{\mathrm{m}1}=r_{\mathrm{m}2}=10$ , $m_0=10\mathrm{kg}$ , $\delta ={60}^{\circ }$ , $r=0.1\mathrm{m}$ , and $l_0=0.5\mathrm{m}$ .

The variations of the coefficients of synchronization ability and the phase differences with the mass ratio $r_{\mathrm{mz}}$ are shown in Figure 4. The coefficients of synchronization ability are always greater than 1, as illustrated in Figure 4(a). This fact demonstrates that the vibrating system can operate in the steady states of self-synchronization. ¹³ The phase difference for single pair of the URs of beams 1 or 2 is zero, the phase difference between the two pairs of the URs is $\pi$ , and all the phase differences keep constant with $r_{\mathrm{mz}}$ value, as illustrated in Figure 4(b). According to the expression of $m_2=M_1/(r_{\mathrm{mz}}-M_1)$ , we have $r_{\mathrm{mz}}<1$ . $m_2$ includes the mass of the pile, which is varied during the different engineering. The fact demonstrates that no matter how much the mass of $m_2$ , the vibrating system can implement the synchronization and the system vibrates in the z-direction all the time.

Figure 4.

The relations between coefficients of synchronous ability, phase difference and mass ratio r_mz.

The variations of the coefficients of synchronization ability and the phase differences with the mass ratio $r_{\mathrm{m}}$ are shown in Figure 5. As illustrated in Figure 5(a), the coefficient of synchronization ability ${\zeta }_2$ always decreases with the increase in $r_{\mathrm{m}}$ value. But when $r_{\mathrm{m}}$ is in the interval of [0, 0.085], the coefficient of synchronization ability $\zeta$ keeps constant while ${\zeta }_1$ decreases from 19.03 to 0; when $r_m>0.085$ , $\zeta$ increases rapidly and ${\zeta }_1$ increases gradually. The coefficient ${\zeta }_1$ is equal to 1 when $r_{\mathrm{m}}=0.0617$ . As illustrated in Figure 5(b), the phase differences, $2{\alpha }_1$ and $2{\alpha }_3$ , are 0 and $\pi$ , respectively. $2{\alpha }_2$ changes from 0 to π when r_m = 0.085. Therefore, $r_{\mathrm{m}}$ should be less than 0.0617. Here, we can assume $r_{\mathrm{m}}=0.05$ .

Figure 5.

The relations between coefficients of synchronous ability, phase difference and mass ratio r_m.

The variations of the coefficients of synchronization ability and the phase differences with the length ratio $r_l(r_{l1}=r_{l2}=r_l)$ are illustrated in Figure 6. As shown in Figure 6, the coefficient $\zeta$ is always greater than 1 and the phase difference is always 180°. The variations of the coefficient ${\zeta }_1$ and ${\zeta }_2$ are two concave curves, which are 0 at $r_l=0.36\mathrm{and}0.71$ , respectively. Furthermore, the phase differences, $2{\alpha }_2$ and $2{\alpha }_3$ , step from 180° to 0°. As illustrated in Figures 1 and 2, the distributions of mass on beams 1 and 2 are large at the two ends and small in the middle, so $r_l\approx 1$ . Therefore, the proposed structure can stratify the requirements: 2α₁ = 180°, $2{\alpha }_2=2{\alpha }_3=0$ , $\zeta >1$ , ${\zeta }_1>1$ , and ${\zeta }_2>1$ .

Figure 6.

The relations between coefficients of synchronous ability, phase difference and displacement ratio r_l.

The variations of the stability parameters $H_1,H_2,H_3,\mathrm{and}{H}_4$ with the structural parameters $r_{\mathrm{mz}}$ , $r_{\mathrm{m}}$ , and $r_l$ are illustrated in Figure 7. As illustrated in Figure 7, the values of $H_1,H_2,H_3,\mathrm{and}{H}_4$ are all greater than zero. This fact demonstrates that the synchronization of the four URs is stable all the time.

Figure 7.

The ability of vibrating system with different amplitudes.

The variations of the phase differences and the rotational speed of hydraulic motors with the capacity ratio are illustrated in Figure 8. As illustrated in Figure 8, the phase differences remain unchanged with the increase in the capacity ratio $\epsilon$ while the rotational speed of the hydraulic motors decreases with the increase in the capacity ratio $\epsilon$ . Hence, if large excitation frequency is needed in the engineering, it can be obtained by decreasing the capacity ratio ε values.

Figure 8.

The relations between phase difference, speed of hydraulic motor and capacity ratio.

Computer simulation

In order to verify the above investigations, computer simulation for the hydraulic motors with different parameters is carried out. Using the fourth-order Runge–Kutta method, we develop the numeric algorithm with the MATLAB software, which is similar to that of the vibrating system with two URs described in Zhao et al. ²¹ The parameters of the system are assumed that $r_{\mathrm{m}}=0.05$ , $\epsilon =0.8$ , ${\eta }_{\mathrm{mv}}=0.95$ , ${\eta }_{\mathrm{bv}}=0.95$ , ${\eta }_{\mathrm{bm}}=0.95$ , and ${\eta }_{\mathrm{mm}}=0.95$ . Block diagram of simulation algorithm is illustrated in Figure 9. Simulation results are shown in Figure 10. As illustrated in Figure 10, during the starting process of the system, the difference of the hydraulic motors results in that the four URs are not synchronous, which excites vibrations in x-, y-, ψ₁-, and ψ₂-directions and the phase differences vary rapidly. With the increase in the rotational speeds for the motors, the synchronous torques increase and take the role of adjusting the phase differences and synchronizing the four URs.^13,18 Finally, the phase differences between URs 1 and 2, 3 and 4, $2{\alpha }_2$ and $2{\alpha }_3$ , stabilize at 15.5° and 4.7°, respectively. The phase difference between the two pairs of URs, $2{\alpha }_1$ , is 185.4°. The rotational speed of synchronization for the four URs is 1120 r/min. The amplitudes of vibration in x-, y-, ψ₁-, and ψ₂-directions are gradually cut down by the dumping. The amplitude of vibration in z-direction rapidly increases to a steady value, 3.2 mm and that in z_r-direction is 0.005 mm. Because the phase differences deviate from their ideal values, $2{\alpha }_1={180}^{\circ }$ and $2{\alpha }_2=2{\alpha }_3=0$ , the four URs also excite steady vibrations of the system in x-, y-, ψ₁-, and ψ₂-directions. But their amplitudes are very small. When time is 6 s, a phase disturbance of 30° is acted on the UR 1, the phase differences quickly return to their steady values.

Figure 9.

Block diagram of simulation algorithm.

Figure 10.

Results of computer simulation for vibrating pile driver with four hydraulic motors rotating in the same directioin: (a)~(f) displacements in x, y, z, z, ψ₁, ψ₂ direction; (g) rotational speeds of the four hydraulic motors; (h) phase differences between every two eccentric rotors.

Conclusion

From the theoretical investigation and numeric analysis given in the above sections, the following remarks can be stressed.

1. A new vibratory pile driver with four hydraulic motors, which can implement self-synchronization, is proposed. The hydraulic system of vibrating pile driver is designed and the mathematical model of the electric motor-pump-hydraulic motor drive system in quasi-steady-state operation is derived.

2. The dimensionless coupling equations of the four URs are deduced by virtue of the modified average method of small parameters. Based on the existence and stability of the zero solution for the dimensionless coupling equations, the synchronization criterion and the stability criterion are derived. The numeric results demonstrate that the synchronization of the four URs is stable all the time.

3. No matter how much the mass of $m_2$ , the vibrating system can implement the synchronization and it vibrates in the z-direction all the time. Furthermore, the coefficient of synchronization ability for single pair of the URs on beam 1 first reduces to zero and then increases again along with the increase in $r_{\mathrm{m}}$ value. In the meantime, that on beam 2 decreases with the increase in $r_{\mathrm{m}}$ value. If the system operates in a steady state of self-synchronization and vibrates only in z-direction, $r_{\mathrm{m}}$ should be less than 0.0617.