Synchronization of the secondary isolation system with a dual-motor excitation

1. Introduction

Synchronization phenomena exists in many aspects of life, such as synchronization in uncoupled neuron system [1], gears [2] and coupled self-sustained electromechanical devices [3]. The phenomenon of synchronization was first discovered by a Dutch physicist Huygens by observing two swinging clocks. After that, Signul, Swedish inventor, has applied for the patent of simplest self-synchronous vibrating machine in 1950 [4]. In 1960s, professor Blekhman of the Soviet Union studied the stability of two synchronously operating exciters by the Poincare-Lyapunov method, and gave the definition of synchronization from the point of view of kinematics and dynamics [5]. On this basis, professor Wen et al proposed the method of small parameter modified average, and solved the synchronization and stability of multiple rotors system [6-8]. Sperling presented a two-plane automatic balancing device for equilibration of rigid-rotor unbalance, and it not only derives dynamic equations for the considered system, but also gives to the discussion of numerical computation findings as well as determines of the synchronous stability conditions according to an analytical approximation [9]. Besides, Balthazar gave some comments on synchronization in pre-resonance and resonance region between DC motors coupling with a flexible strut structure by numerical simulations, and “Sommerfeld effect” of self-synchronization of the system was also analyzed by numerical simulations [10]. Li studied double synchronous states of two motors with horizontal asymmetric structure, and found that the synchronous state can be obtained easily when the system operate in super-resonant state [11]. Synchronization condition and synchronous stability of an elastically coupled rotors in vibration systems are considered by Fang and Hou et al, it is indicated that the synchronous characteristics between rotors are also influenced by stiffness coefficient of coupling springs [12, 13]. Based on active control strategy, Kong implemented ideal synchronization control between the three rotors by employing an adaptive sliding mode algorithm [14, 15]. The above scholars discussed the synchronization theory and synchronization control theory of rotors in vibration system, but theoretical research on the synchronization of rotors in vibrating isolation system is less reported.

At present, with the wide application of vibrating machines, the adverse impacts in their operation process are also gradually appeared. Especially in the synchronous operation for large vibrating screens of offshore drilling platform, some vibration forces produced by the unbalanced rotors operating are transmitted to supported platforms, which can directly lead to platform vibration and generate noise. Meanwhile, some serious accidents are caused such as bolt looseness, fatigue failure of the supported platform and even hazardous for mankind's physical and mental health. In this context, Li proposed a vibrating machine with a two-stage vibration isolation frame and discussed its self-synchronization theory, which found that self-synchronous motion is also achieved when the parameters of vibration system simultaneously satisfy the condition of self-synchronous motion and the stability condition [16]. Anti-resonance machines of a new vibration machinery are proposed by Liu, as well as synchronization and synchronous transmission are studied on considering speed of motors, spring coefficients and geometric parameters of the system [17, 18]. Besides, Liu concerned self-synchronous motion of a dual-mass with two motors vibrating system, and analyzed quantitatively speed of two motors, phase difference between two motors and motion of the system in vibrating direction [19]. Therefore, in present work, taking dynamical model of the secondary isolation system with a dual-motor excitation for example, the synchronous stability will be discussed by the Poincare method, which provide theoretical guidance for designing new types of vibrating isolation machines.

The synchronous characteristics of the secondary isolation system with a dual-motor excitation is discussed in the following sections. The simplified model is described by mathematical formula in Section 2. The stable approximation solutions of vibrating system are obtained by the Laplace transformation method in Section 3. The synchronization equation and stable condition of the vibrating system are obtained through Poincare-Lyapunov method in Section 4. The stable phase difference is numerically analysis through the theoretical results in Section 5. The results of theoretical analysis are verified through computer simulation in Section 6. Finally, the conclusions are summarized in Section 7.

2. Simplified model description

Fig. 1 shows the dynamic model with a dual-motor excitation in the secondary isolation system. The system consists of a rigid vibrating body, a isolation body, springs, unbalanced rotors and motors. The two induction motors are directly fixed installed in vibrating body, which is connected with the isolation body in the vertical direction and foundation in the horizontal direction by the springs ($k_x,k_y,k_{\mathrm{\Psi }}$) [N/m]. The isolation body is directly connected to the foundation by two springs ($k_c$) [N/m]. Two unbalanced rotors actuated by the motors produce the excitation forces to stimulate the motion of the whole system. And the distance between the pivot of two motors and the centroid of the vibrating body is $l$ [m]. ${\beta }_1$ [rad] represents the installed angle of motor 1 between $ox$ and $o{o}_1$, and ${\beta }_2$ [rad] represents the installed angle of motor 2 between $ox$ and $o{o}_2$. Eccentric distance of the unbalanced rotors is denoted by $r$ [m]. The damping coefficients of the springs are $f_x$, $f_y$, $f_c$ and $f_{\mathrm{\Psi }}$ [N·s/m], respectively. $x$, $y$ and $\psi$ are displacement responses of centroid of the vibration body in $x$-, $y$- and $\psi$- direction, respectively. $y_c$ is displacement response of isolation body in $y_c$- direction. ${\phi }_1$ and ${\phi }_2$ are the initial phase of rotor 1 and 2, respectively. $\sigma$ represents the rotation direction of the rotors. The two rotors rotate in the same direction when $\sigma =$ –1, and the two rotors rotate in the opposite direction when $\sigma =$1.

In reference frame $o^{\text{'}}{x}^{\text{'}\text{'}}{y}^{\text{'}\text{'}}$, the center coordinates ${\mathrm{\Phi }}_1^{\text{'}\text{'}}$ and ${\mathrm{}\mathrm{\Phi }}_2^{\text{'}\text{'}}$ of the two unbalanced rotors can be separately expressed as:

Fig. 1The model of the secondary isolation system with a dual-motor excitation: a) dynamic model, b) the reference frame of system

a)

b)

In the reference frame $oxy$, the center coordinates ${\mathrm{\Phi }}_i$ ($i=$ 1, 2) of two rotors can be separately written as:

In this case. The kinetic energy $T$ of the vibration system can be obtained by:

where, $J_m$ is rotational inertia of the vibration body, [Kg∙m²]; $J_{10}$ and $J_{20}$ are rotational inertia of the two rotors, [Kg∙m²].

The potential energy $V$ of the vibration system can be described as:

The dissipation energy $D$ of the whole system can be given by:

Finally, dynamic equations of the system can be deduced by Lagrange equation:

where, in the vibration system, the generalized force matrix $Q$ can be assumed as:

where, $M_{e1}$ and $M_{e2}$ are the electromagnetic torque of induction motors. $R_{e1}$ and $R_{e2}$ are the friction and damping torque of the rotors.

Substituting Eqs. (4), (5), (6) and (8) into Eq. (7), considering $\psi \ll 1$, $m_i\ll m_0$, the dynamic equations of the system can be obtained as follows:

where, $M=m_0+m_1+m_2$, $J_m\approx M{l}_e^2$, $J_{i0}\approx m_i{r}^2$, $i=\mathrm{1,2}$.

3. Stable approximation solutions of the system

In the light of Eq. (9), the vibration equations are coupled in the direction of $y$ and $y_c$. In steady state of the system, the displacements of the system are weakly influenced by the angular acceleration of the rotors, therefore, acceleration $\ddot{{\phi }_1}$ and ${\ddot{\phi }}_2$ closed to zero can be ignored. In the process, the following parameters are introduced as follows:

Submitting Eq. (10) into the first four equations of Eq. (9), the following equations are obtained:

The first two equations in Eq. (11) is no coupling term, which can be directly obtained by solving the second order differential equation. The latter two equations are coupled in directions $y$ and $y_c$, and so the Laplace’s transformation method is employed to solve them under initial conditions $y\left(0\right)=0$, $y_c\left(0\right)=0$, $\dot{y}\left(0\right)=0$, ${\dot{y}}_c\left(0\right)=0$. Therefore, the transfer function of relevant formulas can be obtained by

where:

Assuming $s=i{\omega }_m$, Eq. (12) can be simplified as:

where:

In this case, the approximation displacements of the system in steady state can be rewritten as:

where:

In Eqs. (14) and (16), the parameters (${\mu }_x,{\mu }_{\psi },{\mu }_y,{\mu }_{y_c})$ are the coupling coefficient between the rotors, the vibration body and the isolation body, respectively.

The vibration transmission in this paper is described by amplitude ratio in $y$and$y_c$direction. According to Eqs. (15) and (16), the amplitude of the system is the product of amplitude amplification factor and sine function. Ignoring weakly damped of the system (i.e. $\mathrm{s}\mathrm{i}\mathrm{n}\left({\phi }_j-{\gamma }_y\right)\approx \mathrm{s}\mathrm{i}\mathrm{n}\left({\phi }_j-{\gamma }_{y_c}\right)$, $j=$ 1, 2), ability of the vibration transmission can be described by:

where $\rho$ represents ability of the vibration transmission of the system.

4. Synchronization and stability of the system

4.1. Synchronous condition

The synchronous behavior of the secondary isolation system with a dual-motor excitation can be estimated with the Poincare method, and so the phase angle of the rotors can be defined by:

18

${\phi }_1={\omega }_{m}t+{\alpha }_1,{\phi }_2={\omega }_{m}t+{\alpha }_2.$

Asumming $\alpha$ is phase difference between rotor 1 and 2 i.e.:

19

$\alpha ={\alpha }_1-{\alpha }_2.$

According to Eq. (15), the two-order derivative of $x$, $y$ and $\psi$ with respect to time $t$ can be calculated. Substituting $\ddot{x}$, $\ddot{y}$ and $\ddot{\psi }$ into the last two formulas of Eq. (9), and then integrating and averaging this equation related $t$ over period $T$, respectively, synchronous indexes $P_1$ and $P_2$, i.e., average load torque of the motors, can be obtained by:

20

$P_1={\stackrel{-}{M}}_{e1}-{\stackrel{-}{R}}_{e1}+\frac{1}{2}{m}_1{r}^2{\omega }^2\left\{\begin{array}{l}\sigma {\mu }_x{\eta }_2\sin \left({\alpha }_1-{\alpha }_2+{\gamma }_x\right)-{\mu }_x{\eta }_1\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_x-{\mu }_y{\eta }_1\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_y-{\mu }_{\psi }{\eta }_1{r_l}^2\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_{\psi }\\ -{\mu }_y{\eta }_2\mathrm{s}\mathrm{i}\mathrm{n}({\alpha }_1-{\alpha }_2+{\gamma }_y)+{\mu }_{\psi }{\eta }_2{r_l}^2\mathrm{s}\mathrm{i}\mathrm{n}({\alpha }_1-{\alpha }_2+{\beta }_2-\sigma {\beta }_1+{\gamma }_{\psi })]\end{array}\right\},$

21

$P_2={\stackrel{-}{M}}_{e2}-{\stackrel{-}{R}}_{e2}+\frac{1}{2}{m}_2{r}^2{\omega }^2\left\{\begin{array}{c}-{\mu }_x{\eta }_2\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_x-\sigma {\mu }_x{\eta }_1\sin \left({\alpha }_1-{\alpha }_2-{\gamma }_x\right)+{\mu }_y{\eta }_1\sin \left({\alpha }_1-{\alpha }_2-{\gamma }_y\right)\\ -{\mu }_y{\eta }_2\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_y-{\mu }_{\psi }{\eta }_1{r_l}^2\sin \left({\alpha }_1-{\alpha }_2+{\beta }_2-\sigma {\beta }_1-{\gamma }_{\psi }\right)-{\mu }_{\psi }{r_l}^2{\eta }_2\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_{\psi }\end{array}\right\},$

where, ${\stackrel{-}{M}}_{ei}=\frac{1}{T}{\int }_0^T{M}_{ei}dt$ and ${\stackrel{-}{R}}_{ei}=\frac{1}{T}{\int }_0^T{R}_{ei}dt$, $i=$ 1, 2, which represents the electromagnetic torque and friction torque in the motor shafts, respectively.

Eqs. (20) and (21) can be further rearranged as following:

22

$P_1={\stackrel{-}{M}}_{e1}-{\stackrel{-}{R}}_{e1}+\frac{1}{2}M{r}^2{\omega }^2\left\{{{\eta }_1}^2{W}_{s0}+W_c\sin \left(\alpha +{\theta }_c\right)+W_s\cos \left(\alpha +{\theta }_s\right)\right\},$

23

$P_2={\stackrel{-}{M}}_{e2}-{\stackrel{-}{R}}_{e2}+\frac{1}{2}M{r}^2{\omega }^2\left\{{{\eta }_2}^2{W}_{s0}-W_c\sin \left(\alpha +{\theta }_c\right)+W_s\cos \left(\alpha +{\theta }_s\right)\right\},$

where:

24

$W_c=\sqrt{a_c^2+b_c^2},W_s=\sqrt{a_s^2+b_s^2},$
${\theta }_c=\left\{\begin{array}{l}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\frac{b_c}{a_c},\left(a_c\ge 0\right),\\ \pi +\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\frac{b_c}{a_c},\left(a_c<0\right),\end{array}\right.{\theta }_s=\left\{\begin{array}{l}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}-\frac{b_s}{a_s},\left(a_s\ge 0\right),\\ \pi +\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}-\frac{b_c}{a_c},\left(a_s<0\right),\end{array}\right.$
$W_{s0}=-{\mu }_x\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_x-{\mu }_y\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_y-{\mu }_{\psi }{r_l}^2\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_{\psi },$

$a_c=\sigma {\mu }_x{\eta }_1{\eta }_2\mathrm{c}\mathrm{o}\mathrm{s}{\gamma }_x-{\mu }_y{\eta }_1{\eta }_2\mathrm{c}\mathrm{o}\mathrm{s}{\gamma }_y+{\mu }_{\psi }{\eta }_1{\eta }_2{r_l}^2\mathrm{c}\mathrm{o}\mathrm{s}\left({\beta }_2-\sigma {\beta }_1\right)\mathrm{c}\mathrm{o}\mathrm{s}{\gamma }_{\psi },$
$b_c={\mu }_{\psi }{\eta }_1{\eta }_2{r_l}^2\mathrm{s}\mathrm{i}\mathrm{n}\left({\beta }_2-\sigma {\beta }_1\right)\mathrm{c}\mathrm{o}\mathrm{s}{\gamma }_{\psi },$
$a_s=\sigma {\mu }_x{\eta }_1{\eta }_2\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_x-{\mu }_y{\eta }_1{\eta }_2\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_y+{\mu }_{\psi }{\eta }_1{\eta }_2{r_l}^2\mathrm{c}\mathrm{o}\mathrm{s}\left({\beta }_2-\sigma {\beta }_1\right)\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_{\psi },$
$b_s=-{\mu }_{\psi }{\eta }_1{\eta }_2{r_l}^2\mathrm{s}\mathrm{i}\mathrm{n}\left({\beta }_2-\sigma {\beta }_1\right)\mathrm{s}\mathrm{i}\mathrm{n}{\gamma }_{\psi }.$

In Eq. (24), $W_{s0}$ and $W_s$ are defined as sine coefficients of lagging phase ${\gamma }_i$ ($i=x,y,\psi$), $W_c$ is defined as cosine coefficients of lagging phase ${\gamma }_i.$Therefore, the value of $W_{s0}$ and $W_s$ is far smaller than $W_c$ on account of the small value of lagging phase ${\gamma }_i$.

Therefore, the possible synchronous operation between the rotors can be determined according to Eqs. (22) and (23), i.e.:

25

$({\stackrel{-}{M}}_{e1}+{\stackrel{-}{M}}_{e2})-({\stackrel{-}{R}}_{e1}+{\stackrel{-}{R}}_{e2})+\frac{1}{2}M{r}^2{\omega }^2({{\eta }_1}^2+{{\eta }_2}^2)W_{s0}+M{r}^2{\omega }^2\mathrm{c}\mathrm{o}\mathrm{s}(\alpha +{\theta }_s)W_s=0,$

26

$\left({\stackrel{-}{M}}_{e2}-{\stackrel{-}{M}}_{e1}\right)-\left({\stackrel{-}{R}}_{e2}-{\stackrel{-}{R}}_{e1}\right)+\frac{1}{2}M{r}^2{\omega }^2\left({{\eta }_2}^2-{{\eta }_1}^2\right)W_{s0}=M{r}^2{\omega }^2\mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha +{\theta }_c\right)W_c.$

When the two rotors operate in synchronous state, Eq. (25) can be applied to find the approximation of rated speed $\omega$ of the motors. ${\stackrel{-}{M}}_{e1}+{\stackrel{-}{M}}_{e2}$ is the sum of the electromagnetic torque of the two induction motors; the term of ${\stackrel{-}{R}}_{e1}+{\stackrel{-}{R}}_{e2}$ is the sum of the friction torque of the two rotors; the other items represent the load torque of the two rotors. Therefore, the equation of the vibrating torque between the rotors can be obtained by Eq. (25) when the rotors operate in the steady state. And the value of phase difference $\alpha$ can be calculated by Eq. (26). Considering Eq. (26), the synchronous torque $T_{Capture}$ and the difference of residual torque $T_{Difference}$ can be defined as:

27

$T_{Capture}=M{r}^2{\omega }^2{W}_c,$

28

$T_{Difference}=T_{Residual2}-T_{Residual1},$

where, $T_{Residual1}={\stackrel{-}{M}}_{e1}-{\stackrel{-}{R}}_{e1}+M{r}^2{\omega }^2{{\eta }_1}^2{W}_{s0}/2$, $T_{Residual2}={\stackrel{-}{M}}_{e2}-{\stackrel{-}{R}}_{e2}+M{r}^2{\omega }^2{{\eta }_2}^2{W}_{s0}/2$. The $T_{Captrue}$ is the synchronous torque, also known as the frequency capture torque; the $T_{Difference}$ is the difference between the motor 1 and 2 of residual torque. The $T_{Reasidual1}$ and $T_{Reasidual2}$ are the residual torque in motor 1 and 2, respectively.

Substituting Eqs. (27) and (28) into Eq. (26), the phase difference between the rotors can be determined by:

29

$\alpha =\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\frac{T_{Difference}}{T_{Capture}}-{\theta }_c.$

According to Eq. (29), the phase difference between the rotors is a function related to the parameters $T_{Difference}$, $T_{Capture}$ and ${\theta }_c$.

On the account of $\left|\mathrm{s}\mathrm{i}\mathrm{n}(\alpha +{\theta }_c)\right|\le 1$, the synchronous condition of the system can be expressed by:

30

$T_{Capture}\ge \left|T_{Difference}\right|.$

In the light of Eq. (30), in order to guarantee the synchronous operation between the two rotors, the synchronous torques of the system must be greater than or identical with the absolute value of the difference of residual torque between the two motors.

The coefficient of synchronous ability of the system can be expressed by:

31

$\varsigma =\left|\frac{T_L}{T_{Capture}}\right|,$

where, $T_L$ is maximum of the load torque of the system, i.e., $T_L=\frac{1}{2}M{r}^2{\omega }^2[\left({\eta }_1^2+{\eta }_2^2\right)W_{s0}+2W_s]$. The smaller coefficient of synchronous ability, the easy to implement of synchronous operation. When the system synchronization coefficient is greater than 1, it is difficult to realize synchronous motion for the system. In light of Eq. (31), the coefficient of synchronization ability is related to load torque $T_L$ and synchronous torque $T_{Capture}$, that are the function of ${\eta }_1$, ${\eta }_2$, $r_l$, ${\beta }_1$ and ${\beta }_2$ with damping ratio ${\xi }_i$ ($i=x,y,\psi ,y_c$). As shown in Fig. 2, the coefficient of synchronous ability $\zeta$ is gradually increased with the increase of the installation angle $|{\beta }_2-{\beta }_1|$ when the two rotors rotate in reverse direction (i.e., $\sigma =\text{1}$). And the coefficient of synchronous ability $\zeta$ is the first decreased and then stabilized with the increase of $r_l$, as shown in Fig. 2(a). Therefore, the smaller value of $|{\beta }_2-{\beta }_1|$ and the larger value of $r_l$ are in favor of reversely synchronous operation of the rotors. When the two rotors operate in the same direction (i.e., $\sigma =$ –1), the coefficient of synchronous ability $\zeta$ is the first increased and then the decreased with the increase of $r_l$. And the maximum value of the coefficient of synchronous ability is appeared when $r_l=$ 1.4, as shown in Fig. 2(b).

Fig. 2Coefficients of the synchronous ability when η1= 0.02 and η2= 0.02

a)$\sigma =$ 1

b)$\sigma =$–1

Fig. 3Coefficients of the synchronous ability when η1= 0.02

a)$\sigma =$ 1, $\left|{\beta }_2-{\beta }_1\right|=\pi /6$

b)$\sigma =$–1, $\left|{\beta }_2+{\beta }_1\right|=\pi /3$

However, the fluctuation of the synchronous coefficient is gradually decreased with the increase of $|{\beta }_1+{\beta }_2|$. Therefore, synchronization ability of the system is gradually increased with the increase between two motors mounting distance. From Fig. 3(a, b), the coefficient of synchronous ability of the system is increased with the increase of the mass ratio ${\eta }_2$. In other words, the larger difference of the mass between the two rotors, the more difficult to implement synchronous operation of the system.

5. Numerical computations for theoretical analysis

The stable synchronization behavior of the system can be obtained by Poincare method. According to Eqs. (29) and (31), the phase difference is determined by parameters $T_{Difference}$, $T_{Capture}$ and ${\theta }_c$, that is the function of frequency ratios $n_i\left(i=x,y,\psi ,y_c\right)$, damping ratios ${\xi }_i(i=x,y,\psi ,y_c)$, mass ratios (${\eta }_1,{\eta }_2$), geometrical parameters ($l,{\beta }_1,{\beta }_2$) and rotation direction of the rotors ($\sigma$). However, when the damping coefficients are very small, the synchronization state of the system is less affected by damping ratio. Meanwhile, the system is a far resonance system, and the value of frequency ratios is greater than or equal to 5. Therefore, mass ratios, the rotating direction and geometrical parameters are the important parameters to influence behavior of the proposed system.

5.2. Synchronous state for $\mathit{\sigma }=\mathit{}$1, ${\mathit{\eta }}_1\ne {\mathit{\eta }}_2$,$\mathit{}{\mathit{n}}_{\mathit{x}}={\mathit{n}}_{\mathit{y}}={\mathit{n}}_{\mathit{\psi }}={\mathit{n}}_{\mathit{y}\mathit{c}}=\mathit{}$5

Consider the values of the dimensionless parameters in the system to be $\sigma =1$, ${\eta }_1\ne {\eta }_2$, $n_x=n_y=n_{\psi }=n_{yc}=\text{5}\text{.}$ Because of $\left({\stackrel{-}{M}}_{e1}-{\stackrel{-}{M}}_{e2}\right)-({\stackrel{-}{R}}_{e1}-{\stackrel{-}{R}}_{e2})=0\text{,}$ Eq. (26) can be simplified as $\sin \left(\alpha +{\theta }_c\right)=[\left({\eta }_2^2-{\eta }_1^2\right)W_{s0}/2W_c]$. Therefore, stable phase difference α of the system can be obtained by combining with Eq. (31). According to Fig. 4, it is indicated that the value of the stable phase difference is gradually increased with the increase of ${\beta }_1$, and decreased with the increase of ${\beta }_2$. Comparing Fig. 4 (a) and (b), the value of the stable phase difference is weak influenced by the change of mass ratios${\eta }_2$. Meanwhile, the mounting distance between two motors is not influence on the stable phase difference of the system when two motors rotating in opposite direction.

Fig. 4Stable phase difference when η1≠η2

a)${\beta }_2=\pi /6$, ${\eta }_1=\text{0.02}$, ${\eta }_2=\text{0.03}$

b)${\beta }_2=\pi /6$, ${\eta }_1=\text{0.02}$, ${\eta }_2=\text{0.04}$

c)${\beta }_2=\pi /3$, ${\eta }_1=\text{0.02}$, ${\eta }_2=\text{0.03}$

d)${\beta }_2=\pi /3$, ${\eta }_1=$ 0.02, ${\eta }_2=\text{0.04}$