Effect of phase difference on dynamic characteristics for a synchronous vibrating system with double eccentric rotors

Abstract

Phase difference is an important factor affecting the performances of the synchronous vibrating system driven by the two excited motors. The nonlinear dynamic models of the synchronous vibrating system under the action of the nonlinear elastic force are established. The periodic solutions for the synchronous vibrating system are theoretically derived using the nonlinear dynamic models. The stabilities of periodic solution for the synchronous vibrating system are theoretically analyzed using Jacobi matrix of the amplitude-frequency-characteristic equation. Using Matlab, the amplitude-frequency characteristics are analyzed through the selected parameters. The relations between the phase difference and the amplitude in the synchronous vibrating system are also investigated. Various nonlinear phenomena, such as the jump phenomenon and the multiple-valued periodic solutions, are reproduced using relation between the phase difference and the amplitude. The stable periodic solutions can be obtained by the different initial conditions, using Runge–Kutta method. The effects of the phase difference on the amplitude are presented for the changes of system parameters (including the stiffness of the soil and the damping of the soil, the mass of the eccentric block). The effects of the dynamic characteristics on the phase difference are analyzed through the difference rates of the two excited motors and the initial conditions of the system. It has been shown that the research results can provide a theoretical basis for the research of the synchronous vibrating system.

Introduction

Phase difference has been an important factor in the self-synchronous vibrating system with double rotating eccentric rotors. Phase difference is usually explained that the phase difference of  the rotating eccentric rotors on two excited motors. When the eccentric rotors on two excited motors are the reverse and synchronous rotation, the synchronous operation stability of the self-synchronous vibrating system with double rotating eccentric rotors can be achieved, which can be named as the vibration synchronization in the vibration system. The vibration synchronization can be obtained through the phase synchronization (or speed synchronization), which is explained that the phase difference of the two eccentric blocks is 0 or constant. ¹

In recent years, many models about the vibration synchronization for the self-synchronous vibrating system, such as the linear model with the linear stiffness and the simplified ideal model, have been investigated and found in many studies.^2–4 With the development of the nonlinear vibration theory, the researches on the vibration synchronization of the nonlinear model have also been further developed about the self-synchronous vibrating system. For example, some scholars ⁵ investigated the vibration synchronization characteristics about nonlinear vibration system with piecewise linear stiffness. It had shown from studies^6,7 that the harmonic vibration synchronization about the nonlinear vibration system with the flexible nonlinear stiffness was analyzed using the theory of frequency capture. In summary, all of the researches about the vibration synchronization, whether the linear models^8–10 or the nonlinear models,^11–13 were also about the analysis of the phase difference with zero. So the investigations on the phase difference of the two eccentric rotors have become one of the key issues in the synchronous vibrating system. But the investigations on the relation between the phase difference and the amplitude can be rarely found in many references.^14,15,16 Thus the effect of phase difference on dynamic characteristics for the synchronous vibrating system should be studied.

Firstly, the nonlinear vibration models are established to describe the synchronous vibrating system in this paper. Secondly, the dynamic characteristics of the synchronous vibrating system are investigated to find the relation between the phase difference and the amplitude. Finally, the stabilities of periodic solution for the self-synchronous vibrating system are analyzed.

Mathematical model

The vibrating force on the vertical direction will be generated by the eccentric rotors with the reverse rotation on the double excited motors in the synchronous vibrating system. The dynamic model of the synchronous vibrating pile system is shown in Figure 1, as shown in Figure 1, oxy is the coordinate system of the nonlinear vibrating system, O is the center of the system (it is also the midpoint of the line of the rotating shaft for the eccentric rotors), O₁ and O₂ are the centers of the rotating shafts for the eccentric rotors. Using Lagrange equation, the differential equations of the synchronous vibrating system under the action of the nonlinear elastic force are defined as

$$\begin{array}{l}m\ddot{y}+c\dot{y}+ky\text{-}\epsilon k{y}^3=m_1{r}_1(-{\ddot{\phi }}_1\cos {\phi }_{\text{1}}+{\dot{\phi }}_{\text{1}}{\hspace{0.17em}}^2\sin {\phi }_{\text{1}})-m_{\text{2}}{r}_{\text{2}}({\ddot{\phi }}_{\text{2}}\cos {\phi }_{\text{2}}-{\dot{\phi }}_{\text{2}}{\hspace{0.17em}}^2\sin {\phi }_{\text{2}})\\ J_{0}1{\ddot{\phi }}_{\text{1}}=T_{m1}-T_{f1}-c_1{\dot{\phi }}_{\text{1}}\text{-}{m}_1{r}_1\ddot{y}\cos {\phi }_{\text{1}}\text{-}{m}_1{r}_{1}g\cos {\phi }_{\text{1}}\\ J_{0}2{\ddot{\phi }}_{\text{2}}=T_{m2}-T_{f2}-c_2{\dot{\phi }}_{\text{2}}-m_2{r}_2\ddot{y}\cos {\phi }_{\text{2}}\text{-}{m}_2{r}_{2}g\cos {\phi }_{\text{2}}\end{array}$$

(1)

Figure 1.

Dynamic model of synchronous vibratory system.

In equation (1), $y$ , $\dot{y}$ and $\ddot{y}$ are the vibration displacement, the velocity and the acceleration of the pile of the vibrating pile system on the vertical direction, respectively. m is the total mass of the vibrating pile system and the total mass m is composed of two components, including the mass of the vibratory pile hammer M and the masses of two eccentric blocks on two eccentric rotors (m₁ and m₂), so m can be written as m=M+m₁+m₂. r_i(i=1,2) is the radius of the eccentric block on the eccentric rotor around O_i. ${\phi }_i$ , ${\dot{\phi }}_i$ and ${\ddot{\phi }}_i$ (i=1,2) are the angular phase, the angular velocity and the angular acceleration of the eccentric block on the eccentric rotor around O_i, respectively. c is the damping of the soil on the vibrating pile hammer. $c_i$ (i=1,2) is the rotating damping of the eccentric rotor around O_i. $J_{0i}$ (i=1,2) is the moment of the inertia of the eccentric rotor around O_i. $T_{mi}$ (i=1,2) is the electromagnetic torque of the eccentric rotor on the eccentric motor around O_i. $T_{fi}$ (i=1,2) is the load torque of the eccentric rotor on the eccentric motor around O_i. k is the linear elastic stiffness. The elastic force of the flexible nonlinear spring can be defined as $k(y)=ky-\epsilon k{y}^3$ . ε is the nonlinear coefficient of the soil (which is a small integer) and also the minute quantity. $\epsilon k{y}^3$ is the nonlinear elastic force, which is usually less than ky.

Theoretical analyses

Analytical solution

The angular velocity ${\dot{\phi }}_i$ is generated through the rotation of the eccentric rotor on the eccentric motor around O_i. ${\dot{\phi }}_i$ can be replaced by ${\omega }_i$ (i=1,2) ( ${\omega }_i$ is the angular frequency). When ${\omega }_1={\omega }_2$ , the synchronous operation of the two eccentric motors is presented in the synchronous vibration system. The two angular phases can be defined as ${\phi }_1=\phi +\frac{1}{2}\Delta \phi$ and ${\phi }_2=\phi -\frac{1}{2}\Delta \phi$ . $\phi$ is named as the average angular phase of the two eccentric motors. $\Delta \phi$ is named as the phase difference angle of the eccentric blocks on the two eccentric rotors. $\dot{\phi }$ is named as the average speed of two eccentric blocks. $\dot{\phi }$ may be expressed as $\overline{\omega }$ ( $\overline{\omega }$ is the excited frequency). $\overline{\omega }$ can be defined and expressed as $\overline{\omega }=\frac{{\omega }_1+{\omega }_2}{2}$ . Using the above parameters, the first equation in equation (1) can be rewritten as

$$\ddot{y}+{\omega }_n^{2}y=-2\xi {\omega }_n\dot{y}+\epsilon {\omega }_n{y}^3+F\text{cos}\left(\phi -90°\right)$$

(2)

In equation (2), ${\omega }_n=\sqrt{\frac{k}{m}}$ , $2\xi {\omega }_n=\frac{c}{m}$ and $F=\frac{\left({\displaystyle {\sum }_{i=1}^2{m}_i{r}_i{\dot{\phi }}_i^2}\right)\cos \frac{1}{2}\Delta \phi }{m}$ . In addition, if m₁=m₂=m₀, r₁=r₂=r₀, $F=\frac{2m_0{r}_0{\overline{\omega }}^2\cos \frac{1}{2}\Delta \phi }{m}$ . When the excited frequency is close to the definite range of the first natural frequency in the vibrating pile system, the excited frequency is captured by the first natural frequency, namely, $\overline{\omega }\approx {\omega }_n$ . The stable solution of the synchronous vibrating system can be carried out using multi-scale method. Namely, $2\xi {\omega }_n=2\epsilon u$ (u is damping coefficient of c), equation (2) can be expressed as

$$\ddot{y}+{\omega }_n^{2}y=-2\epsilon u\dot{y}+\epsilon {\omega }_n{y}^3+\epsilon F_0\cos \left[\left({\omega }_n+\epsilon \sigma \right)t-90\circ \right]$$

(3)

In equation (3), $F=\epsilon F_0$ , $\overline{\omega }={\omega }_n+\epsilon \sigma$ . $\epsilon$ is a minute quantity, $\sigma$ is named as the harmonic parameters.

The periodic solutions of equation (3) is deduced using two different time scales T₀ and T₁, namely, $T_0=t,T_1=\epsilon t$ (t is the time). Then the periodic solutions of equation (3) can be assumed as the power series of the minute quantity $\epsilon$ and $\epsilon$ is taken to the first power, namely, $y\left(t,\epsilon \right)=y_0\left(T_0,T_1\right)+\epsilon y_1\left(T_0,T_1\right)$ . The derivatives of y(t) are expressed as $D_n=\frac{\partial }{\partial T_n}$ (n=1,2). The periodic solutions can be substituted into equation (3) that can be transformed as

$$\begin{array}{l}{D}_0^2{y}_0+2\epsilon D_0{D}_1{y}_0+{\epsilon }^2{D}_1^2{y}_0+\epsilon D_0^2{y}_1+2{\epsilon }^2{D}_0{D}_1{y}_1+{\epsilon }^3{D}_1^2{y}_1+{\omega }_n^2{y}_0+\epsilon {\omega }_n^2{y}_1\\ =-2\epsilon u\left(D_0{y}_0+\epsilon D_1{y}_0+\epsilon D_0{y}_1+{\epsilon }^2{D}_1{y}_1\right)+\epsilon {\omega }_n^2{\left(y_0+\epsilon y_1\right)}^3+\epsilon F_0\cos \left[\left({\omega }_n+\epsilon \sigma \right)t-90\circ \right]\end{array}$$

(4)

All the coefficients of $\epsilon$ are equal in equation (4) that can be expressed as

$$\begin{array}{l}{D}_0^2{y}_0+{\omega }_n^2{y}_0=0\\ D_0^2{y}_1+{\omega }_n^2{y}_1=-2D_0{D}_1{y}_0-2u{D}_0{y}_0+{\omega }_n^2{y}_0^3+F_0\cos \left({\omega }_n{T}_0+\sigma T_1-90\circ \right)\end{array}$$

(5)

The general solution of $y_0$ in the first equation of equation (5) can be written as $y_0=C\left(T_1\right)e^{j{\omega }_n{T}_0}+\overline{C}\left(T_1\right)e^{-j{\omega }_n{T}_0}$ . $C$ and $\overline{C}$ are named as the conjugate complex number. The second equation of equation (5) is substituted by $y_0$ with the conjugate complex number. The second equation of equation (5) can be transformed as

$$\begin{array}{l}{D}_0^2{y}_1+{\omega }_n^2{y}_1=-\left[2j{\omega }_n\left(\dot{C}+uC\right)-3{\omega }_n^2{C}^2\overline{C}\right]e^{j{\omega }_n{T}_0}-\left[-2j{\omega }_n(\dot{C}+u\overline{C})-3{\omega }_n^{2}C{\overline{C}}^2\right]e^{-j{\omega }_n{T}_0}\\ +{\omega }_n^2{C}^3{e}^{j3{\omega }_n{T}_0}+{\omega }_n^2{\overline{C}}^3{e}^{\text{-}j3{\omega }_n{T}_0}+\frac{1}{2}{F}_0{e}^{j(\sigma T_1-90\circ )}{e}^{j{\omega }_n{T}_0}+\frac{1}{2}{F}_0{e}^{-j(\sigma T_1-90\circ )}{e}^{-j{\omega }_n{T}_0}\end{array}$$

(6)

The secular term in equation (6) should be removed and can be defined as

$$2j{\omega }_n(\dot{C}+uC)-3{\omega }_n^2{C}^2\overline{C}-\frac{1}{2}{F}_0{e}^{j(\sigma T_1-90\circ )}=0$$

(7)

In equation (7), $C$ and $\overline{C}$ in $y_0$ is a function of T₁ and can be expressed as exponential function. So they can be written as $C=\frac{1}{2}\alpha (T_1)e^{j\beta (T_1)}$ and $\overline{C}=\frac{1}{2}\alpha (T_1)e^{-j\beta (T_1)}$ . $\alpha (T_1)$ and $\beta (T_1)$ is also the function of T₁. So $C$ and $\overline{C}$ can also be expressed as follows

$$\begin{array}{c}C=\frac{1}{2}\alpha \left(T_1\right)\cos \beta \left(T_1\right)+\frac{1}{2}j\alpha \left(T_1\right)\sin \beta \left(T_1\right)\\ \overline{C}=\frac{1}{2}\alpha \left(T_1\right)\cos \beta \left(T_1\right)-\frac{1}{2}j\alpha \left(T_1\right)\sin \beta \left(T_1\right)\end{array}$$

(8)

Equation (8) is carried into equation (7) and can be written as

$$\begin{array}{c}j{\alpha }^{\prime }\cos \beta -j\alpha {\beta }^{\prime }\sin \beta -{\alpha }^{\prime }\sin \beta -\alpha {\beta }^{\prime }\cos \beta +ju\alpha \cos \beta -u\alpha \sin \beta -\frac{3{\omega }_n^2}{8}{\alpha }^{3}c\text{os}\beta \\ -j\frac{3{\omega }_n^2}{8}{\alpha }^3\sin \beta -\frac{1}{2{\omega }_n}{F}_0\cos \left(\sigma T_1-90\circ \right)-j\frac{1}{2{\omega }_n}{F}_0\sin \left(\sigma T_1-90\circ \right)=0\end{array}$$

(9)

In equation (9), ${\alpha }^{\prime }$ is the time derivative of $\alpha$ and ${\beta }^{\prime }$ is the time derivative of $\beta$ . The coefficients of real and imaginary parts on both sides of the equation are equally in equation (9), respectively. So equation (9) can be expressed as

$$\begin{array}{c}{\alpha }^{\prime }\cos \beta -\alpha {\beta }^{\prime }\sin \beta =-u\alpha \cos \beta +\frac{3{\omega }_n}{8}{\alpha }^3\sin \beta +\frac{1}{2{\omega }_n}{F}_0\sin \left(\sigma T_1-90\circ \right)\\ -{\alpha }^{\prime }\sin \beta -\alpha {\beta }^{\prime }\cos \beta =u\alpha \sin \beta +\frac{3{\omega }_n}{8}{\alpha }^3\cos \beta +\frac{1}{2{\omega }_n}{F}_0\cos \left(\sigma T_1-90\circ \right)\end{array}$$

(10)

Equation (10) should be transformed into an autonomous system (T₁ is not included in the system), so the first equation is multiplied by $\sin \beta$ and the second equation is multiplied by $\cos \beta$ in equation (10), then they add. The first equation is multiplied by $\cos \beta$ and the second equation is multiplied by $\sin \beta$ , then they subtract. Equation (10) can be rewritten as

$$\begin{array}{l}\alpha {\gamma }^{\prime }=\alpha \sigma +\frac{3{\omega }_n}{8}{\alpha }^3+\frac{F_0}{2{\omega }_n}\cos \gamma \\ {\alpha }^{\prime }=-u\alpha +\frac{F_0}{2{\omega }_n}\sin \gamma \end{array}$$

(11)

In equation (11), ${\gamma }^{\prime }=\sigma -{\beta }^{\prime }$ . When the system is running in a stable state, ${\alpha }^{\prime }={\beta }^{\prime }=0$ . The square of the first equation and the square of the second equation are added in equation (11). Equation (11) can be transformed as

$${\left(\xi {\omega }_n\right)}^2+{\left(\overline{\omega }-{\omega }_n+\frac{3\epsilon {\omega }_n}{8}{\alpha }^2\right)}^2{\alpha }^2=\frac{F^2}{4{\omega }_n^2}$$

(12)

In equation (12), $\tan \gamma =\frac{-u}{\sigma +\frac{3{\omega }_0^2}{8{\omega }_n}{\alpha }^2}$ . $\tan \gamma$ is obtained by equation (11), when the second equation is divided by the first equation in equation (11). It has been shown that the amplitude-frequency characteristic of the synchronous vibrating system has been expressed in equation (12). The periodic solution of equation (1) in the synchronous vibrating system can be sorted and expressed as

$$\begin{array}{l}{y}_0=C\left(T_1\right)e^{j{\omega }_n{T}_0}+\overline{C}\left(T_1\right)e^{-j{\omega }_n{T}_0}=\frac{1}{2}\alpha e^{j\beta }{e}^{j{\omega }_n{T}_0}+\frac{1}{2}\alpha e^{-j\beta }{e}^{-j{\omega }_n{T}_0}=\alpha \cos \left({\omega }_n{T}_0+\sigma T_1-\gamma -90\circ \right)\end{array}$$

(13)

In equation (13), $\beta =\sigma T_1-\gamma -90\circ$ . So the periodic solution of equation (1) in the synchronous vibrating system can be rewritten as

$$y=\alpha \sin \left(\overline{\omega }t-\gamma \right)$$

(14)

In equation (14), $\alpha$ and $\gamma$ in are determined by the parameters of equation (12).

Amplitude-frequency characteristic

The certain parameters in the synchronous vibrating system may be selected and be represented in Table 1. The amplitude-frequency characteristic can be analyzed through the selected parameters in equation (12). Then using Matlab, the curve of the amplitude-frequency characteristic is shown in Figure 2.

Figure 2.

Curve of amplitude-frequency characteristic.

Table 1.

The certain parameters in the synchronous vibrating system.

m (kg)	k (N/m)	c (N·s/m)	$\epsilon$	$\Delta \phi$	m0 (kg)	r0 (m)
10	40	0.4	0.4	$\text{π}/6$	2.5	0.08

The effect of phase difference on dynamic characteristics

Stabilities of periodic solution

A stable periodic solution of the synchronous vibrating system in equation (14) can be presented for the analysis of $\alpha$ and $\gamma$ in equation (12). The stability of the periodic solution in the synchronous vibrating system is theoretically analyzed through the stability of $\alpha$ and $\gamma$ in equation (11). The stability of $\alpha$ and $\gamma$ in equation (11) can be analyzed and transformed as the following

$$\{\begin{array}{l}P=\alpha \cos \gamma \\ Q=\alpha \sin \gamma \end{array}$$

(15)

The derivatives of equation (15) are substituted into equation (11) and can be re-expressed as

$$\{\begin{array}{l}\frac{dP}{dt}\cos \gamma +\frac{dQ}{dt}\sin \gamma =-u\alpha +\frac{F_0}{2{\omega }_n}\sin \gamma \\ -\frac{dP}{dt}\sin \gamma +\frac{dQ}{dt}\cos \gamma =\alpha \sigma +\frac{3{\omega }_n}{8}{\alpha }^3+\frac{F_0}{2{\omega }_n}\cos \gamma \end{array}$$

(16)

In equation (16), the first equation is multiplied by $\cos \gamma$ and the second equation is multiplied by $\sin \gamma$ , then they are subtracted. Similarly, the first equation is multiplied by $\sin \gamma$ and the second equation is multiplied by $\cos \gamma$ , then they are added. Equation (16) can be transformed as

$$\{\begin{array}{l}\frac{dP}{dt}=-uP-\sigma Q-\frac{3{\omega }_n}{8}Q\left(P^2+Q^2\right)\\ \frac{dQ}{dt}=\sigma P-uQ+\frac{3{\omega }_n}{8}P\left(P^2+Q^2\right)+\frac{F_0}{2{\omega }_n}\end{array}$$

(17)

The stabilities of $\alpha$ and $\gamma$ are theoretically analyzed using the eigenvalues of Jacobi matrix for equation (11). The equations of $\alpha$ and $\gamma$ in equation (11) have been transformed as the equations of P and Q in equation (17), thus the stabilities of periodic solution in the synchronous vibrating system may be analyzed using Jacobi matrix of equation (17). As shown in equation (12), at least there is a stable periodic solution in equation (11). $\alpha$ and $\gamma$ of a stable periodic solution are named as ${\alpha }_0$ and ${\gamma }_0$ . ${\alpha }_0$ and ${\gamma }_0$ can be obtained through equation (12). So the corresponding P and Q can be obtained and named as P₀ and Q₀ in equation (17), namely, $P_0^2+Q_0^2={\alpha }_0^2$ . The Jacobi matrix of equation (17) at this point (P₀, Q₀) can be written as

$$\left[\begin{array}{cc}-u-\frac{3{\omega }_n}{4}{Q}_0{P}_0& -\sigma -\frac{3{\omega }_n}{8}\left(P_0^2+3Q_0^2\right)\\ \sigma +\frac{3{\omega }_n}{8}\left(3P_0^2+Q_0^2\right)& -u+\frac{3{\omega }_n}{4}{Q}_0{P}_0\end{array}\right]$$

(18)

The eigenvalues equation of equation (18) can be defined and simplified as

$${\left(\lambda +\frac{\xi {\omega }_n}{\epsilon }\right)}^2=-\frac{1}{{\epsilon }^2}\left[\overline{\omega }-\left({\omega }_n-\frac{3\epsilon {\omega }_n}{8}{a}_0^2\right)\right]\left[\overline{\omega }-\left({\omega }_n-\frac{9\epsilon {\omega }_n}{8}{a}_0^2\right)\right]$$

(19)

The stabilities of the periodic solution in the synchronous vibrating system can be theoretically explained as the following three points

When ${\omega }_n-\frac{9\epsilon {\omega }_n}{8}{a}_0^2<\overline{\omega }<{\omega }_n-\frac{3\epsilon {\omega }_n}{8}{a}_0^2$ , the eigenvalues $\lambda$ of equation (18) can be written as ${\lambda }_1=-\frac{\xi {\omega }_n}{\epsilon }+\frac{1}{\epsilon }\sqrt{-\left[\overline{\omega }-\left({\omega }_n-\frac{3\epsilon {\omega }_n}{8{\omega }_n}{a}_0^2\right)\right]\left[\overline{\omega }-\left({\omega }_n-\frac{9\epsilon {\omega }_n}{8}{a}_0^2\right)\right]},$ ${\lambda }_2=-\frac{\xi {\omega }_n}{\epsilon }-\frac{1}{\epsilon }\sqrt{-\left[\overline{\omega }-\left({\omega }_n-\frac{3\epsilon {\omega }_n}{8}{a}_0^2\right)\right]\left[\overline{\omega }-\left({\omega }_n-\frac{9\epsilon {\omega }_n}{8}{a}_0^2\right)\right]}$

Obviously, ${\lambda }_2<0$ . If $\xi {\omega }_n>\sqrt{-\left[\overline{\omega }-\left({\omega }_n-\frac{3\epsilon {\omega }_n}{8}{a}_0^2\right)\right]\left[\overline{\omega }-\left({\omega }_n-\frac{9\epsilon {\omega }_n}{8}{a}_0^2\right)\right]}$ (namely, ${\lambda }_1<0$ ), the periodic solution at (P_0,Q₀) should be asymptotically stable. If $\xi {\omega }_n<\sqrt{-\left[\overline{\omega }-\left({\omega }_n-\frac{3\epsilon {\omega }_n}{8}{a}_0^2\right)\right]\left[\overline{\omega }-\left({\omega }_n-\frac{9\epsilon {\omega }_n}{8}{a}_0^2\right)\right]}$ (namely, ${\lambda }_1>0$ ), the periodic solution at (P_0,Q₀) should be not stable. If $\xi {\omega }_n=\sqrt{-\left[\overline{\omega }-\left({\omega }_n-\frac{3\epsilon {\omega }_n}{8}{a}_0^2\right)\right]\left[\overline{\omega }-\left({\omega }_n-\frac{9\epsilon {\omega }_n}{8}{a}_0^2\right)\right]}$ (namely, ${\lambda }_1=0$ ), the periodic solution cannot be determined.

b. When $\overline{\omega }={\omega }_n-\frac{9\epsilon {\omega }_n}{8}{a}_0^2$ or $\overline{\omega }={\omega }_n-\frac{3\epsilon {\omega }_n}{8}{a}_0^2$ , the eigenvalues $\lambda$ of Jacobi matrix in equation (17) can be expressed as $\lambda =-\frac{\xi {\omega }_n}{\epsilon }$ . The eigenvalues $\lambda$ can be negative, so the periodic solution can be asymptotically stable.

c. When $\overline{\omega }>{\omega }_n-\frac{3\epsilon {\omega }_n}{8}{a}_0^2$ or $\overline{\omega }<{\omega }_n-\frac{9\epsilon {\omega }_n}{8}{a}_0^2$ , the eigenvalues of Jacobi matrix in equation (17) are two conjugate complex roots, namely, ${\lambda }_{1,2}=-\frac{\xi {\omega }_n}{\epsilon }\pm i\sqrt{-\frac{1}{{\epsilon }^2}\left[\overline{\omega }-\left({\omega }_n-\frac{3\epsilon {\omega }_n}{8}{a}_0^2\right)\right]\left[\overline{\omega }-\left({\omega }_n-\frac{9\epsilon {\omega }_n}{8}{a}_0^2\right)\right]}$ . The real parts of the eigenvalues are negative, so the periodic solution should be asymptotically stable.

Phase difference-amplitude characteristic

When the excited frequency is close to the definite range of the first natural frequency in the vibrating system (namely, $\overline{\omega }\approx {\omega }_n$ ), the excited frequency is captured by the natural frequency of the system. When the first natural frequency is 2 rad/s and $\overline{\omega }$ =1.9 rad/s, using the system parameters in the second section, the relation between the phase difference and the amplitude in equation (12) can be obtained and shown in Figure 3.

Figure 3.

Relation between the phase difference and amplitude. (a) Under nonlinear conditions and (b) Under linear and nonlinear conditions.

As shown in Figure 3(a), the maximum amplitude is presented for the phase difference $\Delta \phi$ at 0 or $2\text{π}$ , and then the minimum amplitude (namely, the amplitude is 0) is presented for the phase difference $\Delta \phi$ at $\text{π}$ . The dotted line represents the unstable solution, which includes the two segments of cd and hi. Because of the nonlinear factors of the vibrating system, the multiple-valued amplitudes are displayed in the segments of be and gj in Figure 3(a). When the phase difference $\Delta \phi$ starts from zero and is continuously increased, the amplitude (from a to c, from c to e, from e to f) is slowly be reduced in Figure 3(a). But only the amplitude from c to e is reduced sharply, the spontaneous jump phenomenon is occurred at such a time, and the jump phenomenon is also named as amplitude mutation. When the phase difference $\Delta \alpha$ starts from $\text{π}$ and is continuously increased, the amplitude (from f to h, from h to j, from j to k) is continuously increased in Figure 3(a). But only the amplitude from h to j is continuously increased and the jump phenomenon is occurred in the synchronous vibrating system. Similarly, when the phase difference $\Delta \alpha$ starts from 2π and is continuously reduced, the amplitude is from k to i, from i to g, from g to f, and the amplitude is constantly reduced in Figure 3(a). But the amplitude from i to g is dramatically reduced, the jump phenomenon is occurred at such a time. When the phase difference $\Delta \phi$ starts from π and is continuously reduced, the amplitude is from f to d, from d to b, from b to a, and is continuously increased in Figure 3(a). When the amplitude from d to b is constantly increased, the jump phenomenon is occurred at such a time.

As shown in Figure 3(a), when the phase difference $\Delta \phi$ is at two intervals of cd and hi, the multi-value amplitudes are obtained at each phase difference. Among them, the dotted lines at two intervals of cd and hi represent the regions of the unstable solution in the synchronous vibrating system, which cannot be realized in any experiment. The other two of the multi-value amplitudes are the amplitudes of the stable solutions, both of which can be achieved in simulation and experiment. The phase difference $\Delta \phi$ with two stable solutions can be found at two intervals of hi and cd, which is the characteristic of the nonlinear system. In addition, the stable solution in the linear system is independent of the initial conditions, but the stable solution in a nonlinear system is determined by the initial conditions. Thus, one of the two stable solutions for the synchronous vibrating system can be obtained through the determination of the initial conditions.

When the nonlinear coefficient ε of the soil is 0 and removed in equation (12), the linear system can be displayed in Figure 3(b). As shown in Figure 3(b), “Linear” in Figure 3(b) represents the linear system, and “Nonlinear” represents the nonlinear system. The amplitude in the linear system is obviously decreased from the maximum amplitude, but the maximum amplitude in the nonlinear system is dropped slowly at these two intervals ab hi and kj, which is named as the stable regions of the maximum amplitude. The stable regions of the maximum amplitude in the nonlinear system are more stable than in the linear system, which are very beneficial to the vibrating machine.

The effect of phase difference on stability of periodic solution

Based on the theoretical analysis about the stability of the periodic solution in the first section, the stability of the periodic solution can also be determined using the relation between the phase difference and the amplitude. As shown in Figure 3(a), these three points on the region of the multiple-valued amplitudes can be found for F₁=(2.22, 0.16), F₂=(2.22, 0.48), F₃=(2.22, 0.61). F₂ is on the dotted line and should be unstable point. The periodic solution at F₂ is unstable solution. So F₂ cannot be achieved in any experiment. F₁ and F₃ should be stable points and the periodic solution at F₁ and F₃ are stable solution in Figure 3(a). The selected system parameters in the synchronous vibrating system are substituted to equation (19). Equation (19) can be written as

$${\left(\lambda +0.04\right)}^2=-4\left(-0.1+0.3a_0^2\right)\left(-0.1+0.9a_0^2\right)$$

(20)

In equation (20), when F₁=(2.22, 0.16), the phase difference $\Delta \phi$ is 2.22 rad and the amplitude $a_0$ of the stable solution is 0.16 m, namely, $\Delta \phi$ =2.22 rad, $a_0$ =0.16 m. $a_0$ is substituted into equation (20), then the eigenvalue $\lambda$ can be obtained and written as $\lambda =-0.04\hspace{0.17em}\pm \hspace{0.17em}i\sqrt{0.028}$ . It is shown that the real part of the eigenvalue $\lambda$ should be negative, so the periodic solution at F₁ should be stable in the synchronous vibrating system. Namely, the stable solution can be obtained at F₁. Similarly, When F₂=(2.22, 0.48), $\Delta \phi$ =2.22 rad and $a_0$ =0.48 m. The eigenvalue $\lambda$ can be obtained and written as $\lambda =0.075$ . Obviously, the real part of the eigenvalues $\lambda$ should be positive, so the unstable solution can be obtained at F₂. When F₃=(2.22, 0.61) (namely, $\mathrm{ \Delta }\phi$ =2.22 rad and $a_0$ =0.61 m), $\lambda =-0.04\hspace{0.17em}\pm \hspace{0.17em}i\sqrt{0.011}$ . Obviously, the real part of the eigenvalues should be negative, so the stable solution can be obtained at F₃. Thus the stability of the periodic solution can be determined using the relation between the phase difference and the amplitude.

The effect of the phase difference on the stability of the periodic solution can be obtained using the phase planes and the waveforms of $\alpha$ and $\gamma$ . These points (F₁, F₂, F₃, d (1.74,0.34) and h (4.54,0.34)) are selected as the key points in Figure 3(a). Using MATLAB/Simulink, the phase planes and the waveforms of $\alpha$ and $\gamma$ in equation (11) can be obtained as shown in Figures 4 to 7. When $\Delta \phi$ = 1.74 rad and $\Delta \phi$ =4.54 rad, the phase planes and the waveforms of $\alpha$ and $\gamma$ are shown in Figures 4 and 5, respectively. As shown in Figures 4 and 5, both the amplitudes and the phases of the periodic solution tend to be stable after an irregular movement. The amplitudes at d and h are finally stable at around 0.66 m. It is shown that the amplitude is constantly increased and immediately moved to b(1.74,0.66) and j(4.54,0.66), respectively. When the phase difference is at d(1.74,0.34) and h(4.54,0.34), the amplitude is constantly increased and moved to b(1.74, 0.66) and j(4.54,0.66), respectively. Now the jump phenomenon is occurred in the synchronous vibrating system. The amplitude on the region of the multiple-valued amplitudes can be obtained through the determination of the initial conditions.

Figure 4.

The phase plane and the waveform at $\Delta \phi =1.74$ rad.

Figure 5.

The phase plane and the waveform at $\Delta \phi =4.54$ rad.

As shown in Figure 3(a), when the phase difference $\Delta \phi$ is 2.22 rad, the three points on the region of the multiple-valued amplitudes can be found, including F₁, F₂ and F₃. When $\Delta \phi$ = 2.22 rad, the phase planes and the waveforms of $\alpha$ and $\gamma$ under the different initial conditions are shown in Figures 6 and 7. As shown in Figure 6, when the initial amplitude is zero (namely, ${\alpha }_{t=0}=0$ ) and the initial phase is also zero (namely, ${\gamma }_{t=0}=0$ ), both the amplitudes and the phases of the periodic solution tend to be stable after an irregular movement. Finally, the amplitude is stable at about 0.16 m and the phase is stable at 0.2 rad, namely, the amplitude is stable at F₁(2.22, 0.16) in Figure 3(a). Similarly, As shown in Figure 7, when the initial amplitude ${\alpha }_{t=0}$ is 0.5 m and the initial phase ${\gamma }_{t=0}$ is 1.57 rad, the amplitude is finally stable at about 0.61 m and the phase is finally stable at 2.25 rad, namely, the stable solution is stable at F₃(2.22, 0.61) in Figure 3(a). As shown in Figures 6 and 7, F₁ and F₃ in Figure 3(a) are with the stable periodic solution. It has been shown that the stable periodic solution on the region of the multiple-valued amplitudes can be presented through the choices of the initial conditions. Thus the stable solution with the large amplitude can be obtained by choosing the appropriate initial conditions.

Figure 6.

The phase plane and the waveform at $\Delta \phi =2.22$ rad, ${\alpha }_{t=0}=0$ , ${\gamma }_{t=0}=0$ .

Figure 7.

The phase plane and the waveform at $\Delta \phi =2.22$ rad, ${\alpha }_{t=0}=0.5\text{\hspace{0.17em}m}$ , ${\gamma }_{t=0}=1.57\mathrm{rad}$ .

The effect of phase difference on the amplitude

When the system parameters are changed in the synchronous vibrating system, the relation between the phase difference and the amplitude is shown in Figure 8. As shown in Figure 8, the maximum amplitude at 0 and 2π can be increased with the decrease of the system parameters (such as the soil damping c and the nonlinear elastic coefficient ε of soil). The maximum amplitude can also be increased through the increase of the linear stiffness $k$ and the mass of the eccentric block. The range of the phase difference at the region of the unstable solution is narrowed through the increase of the system parameters (such as the soil damping c, the nonlinear elastic coefficient $\epsilon$ , the mass of the eccentric block) and the decrease of linear stiffness $k$ . So it has been shown in Figure 8 that the range of the phase difference at the region of the unstable solution can be narrowed and the maximum amplitude can be increased, which can be obtained through the appropriate chance of the system parameters.

Figure 8.

Relation between the phase difference and amplitude under the change of parameters.

The effect of dynamic characteristics on phase difference

The synchronous states of the double eccentric rotors are influenced by the dynamical characteristics of the synchronous vibrating system. Moreover, the dynamical characteristics of the synchronous vibrating system also affect the synchronous state of the double eccentric rotors. So the effect of dynamic characteristics on phase difference should be analyzed as the following.

The last two equations of equation (1) are named as the rotation equation of the double eccentric rotors; they can reflect the effect of dynamic characteristics on phase difference. The simulation is performed using Matlab/Simulink and the responses of the phase difference and dynamic characteristics in the model of the self-synchronous vibrating pile system have been obtained using the combination of equation (1) with the electromagnetic torque equation and the rotor motion equations about the motor. From the point of view of simulation analysis, when the difference rate of the two excited motors is 0 and the initial conditions of the system are 0, the vibration displacement and the response of phase difference are obtained as shown in Figure 9. As shown in Figure 9(a), the vibration displacement tends to be stable and does periodic motion. The displacement of the periodic motion is eventually stabilized at about 0.68 m. As shown in Figure 9(b), when the initial phase difference is 0, the phase difference are always 0.

Figure 9.

Parameters simulation of the system under ideal conditions. (a) The initial displacement (0 m) and (b) The initial phase difference (0 rad).

When the initial displacement of vibration system is −0.02 m, the vibration displacement of the system is shown in Figure 10(a). The vibration displacement tends to be stable and do periodic motion. Finally, the displacement of the periodic motion is eventually stabilized at about 0.68 m, after a significant periodic movement. But the response of the phase difference is the same as the response in Figure 9(b); the phase difference is still 0 and it has no change. It has been shown that the changes of the initial displacement cannot affect the phase difference.

Figure 10.

The system parameter response in different initial conditions. (a) The initial displacement (−0.02 m) and (b) The initial phase difference (1 rad).

When the initial phase difference of vibration system is 1 rad, the response of the phase difference is shown in Figure 10(b). When the initial phase difference of vibration system is π rad, the vibration displacement of the system and the response of phase difference are obtained as shown in Figure 11. As shown in Figures 10(b) and 11, as a result of some differences about the initial phases between double eccentric rotors, the phase differences are all experienced a gentle transition process, and then, high fluctuations of the phase difference are shown; finally, the phase difference is stable at 0 or π rad. When the phase difference is finally stable at 0 in Figure 10(b), the diagram of vibration displacement does not change, which is basically the same as vibration displacement in Figure 9(a). When the phase difference is finally stable at π rad, as shown in the second small graphs of Figure 11, the vibration displacement does the periodic motion in a very small range; finally, the amplitude of the periodic motion is eventually stabilized at about 0. As a result of no vibration displacement, this is not expected in the engineering practice.

Figure 11.

The system parameter response in different initial conditions.

As shown in Figure 3(a), when the phase difference is at 0 or 2π, the maximum amplitude can be obtained at 0 or 2π. But when the phase difference is at π rad, the amplitude is 0. As shown in Figures 9 to 11, when the phase difference is eventually stabilized at 0, the maximum amplitudes can be obtained. When the phase difference is eventually stabilized at π rad, the amplitude of the periodic motion is very small or even 0. This is consistent with the theoretical conclusions in Figure 3(a).

When the difference rates of two excited motors producing the excited force are in a certain range, the synchronous state of the double eccentric rotors can still be realized by itself. The phase difference and the phase plane of the phase difference and the rotational speed difference are shown in Figure 12. As shown in Figure 12, when the difference rates of two excited motors (including the difference of the eccentric mass distance in exciting forces, the different initial conditions about the motors parameters, the difference of the motors parameters) are in a certain range, the phase difference can be stable at 0 (or 2π) and π. The phase plane graph about the phase difference has a limit cycle. It has been shown that the phase difference is stable and the synchronous state of the double eccentric rotors can be achieved. When the phase difference can be stable at 0 or 2π (namely, Figure 12(a) and (b)), the synchronous state of the double eccentric rotors and the synchronous stability operation of the self-synchronous vibrating system can be obtained to achieve the large amplitude of the vibration. When the phase difference can be stable at π rad (namely, Figure 12(c)), the synchronous state of the double eccentric rotors can also be achieved. But the minimum amplitude or even the amplitude of 0 is obtained, which means that there is no vibration in the synchronous vibrating system. So the phase difference can be stable at π rad, which should be avoided in the engineering practice.

Figure 12.

Simulation of the system in the difference of the motors parameters. (a) In the difference of the eccentric mass distance in exciting forces. (b) In the different initial conditions about the motors parameters. and (c) In the difference of the motors parameters.

Conclusion

Here, the stabilities of the periodic solutions in the synchronous vibrating system with double rotating eccentric rotors have been theoretically derived. The stability of the periodic solutions has been discussed in theory and also determined using the relation between the phase difference and the amplitude.

Phase difference-amplitude characteristic has been presented for the analysis of the relation between the phase difference and the amplitude, which is induced by the equation of amplitude-frequency characteristic. It has been shown that three periodic solutions can be obtained at the regions of the phase difference with the multiple-valued amplitudes. Two periodic solutions among the three periodic solutions are stable, while the other is unstable. In addition, the jump phenomenon is occurred at the critical point of the multi-valued amplitude region. Then the stable solution with the large amplitude can be obtained through the choices of the initial conditions.

The amplitude can be improved through the appropriate chance of the system parameters. For example, the maximum amplitude can be increased through the decrease of the system parameters (such as the soil damping c and the nonlinear elastic coefficient ε of soil) and the increases of the stiffness of the soil.

When the difference rates of the two excited motors and the initial conditions of the system are in a certain range, the synchronous state of the double eccentric rotors can still be realized by itself. The changes of the initial displacement cannot affect the phase difference. But some differences about the initial phases can affect the phase difference. When the phase difference can be stable at 0 or 2π, the synchronous state of the double eccentric rotors and the synchronous stability operation of the self-synchronous vibrating system can be obtained to achieve the large amplitude of the vibration. When the phase difference can be stable at π rad, the synchronous state of the double eccentric rotors can be obtained to achieve the minimum amplitude or even the amplitude of 0, which should be avoided because of no vibration in the synchronous vibrating system.