Investigation for synchronization of two co-rotating rotors installed with nonlinear springs in a non-resonance system

Abstract

To avoid anti-phase synchronization for two co-rotating rotors system that occurs so that exciting force generated by the vibrating system is very small, a mechanical model of two co-rotating rotors installed with nonlinear springs is proposed to implement synchronization in a non-resonance system. The dynamic equations of the system are first built up by using Lagrange's equation. Second, an analytical approach, the average method of modified small parameters, is employed to study the synchronization characteristics of the vibrating system, the non-dimensional coupling equations of two motors are deduced, synchronization problem is converted to that of existence and stability of zero solution for the non-dimensional coupling equations of angular velocity. It is indicated that the synchronous torque of two motors coupled with nonlinear springs in synchronous state must be greater than or equal to the difference of their residual torque. Then, in light of the Routh–Hurwitz criterion, the synchronous criterion of the vibrating system is obtained. Obviously, it is demonstrated that the synchronous state and the stability criterion of the system are influenced by the structural parameters of the coupling unit, coupling coefficients and the positional parameters of two rotors, and so on. Especially, there are clearances in between two nonlinear serial springs, which result in synchronization of the vibrating system that lies in an uncertain state. At last, computer simulations in agreement with the numerical results verify the correctness of the theoretical results for solving the steady phase difference between two rotors. It is demonstrated that adjusting the value of the coupling spring stiffness can make phase difference close to zero to meet the requirements of the strongly exciting force in engineering.

Introduction

The synchronization problem is one of the most fascinating nonlinear phenomena, which are widely applied to vibration utilization engineering everywhere in the real world, such as vibrating conveyers, centrifugal machines, vibrating screen and so on. The most representational synchronization is self-synchronization of unbalanced rotors. In 1665, Huygens, a Dutch researcher, was the first one to discover the synchronization of oscillating pendulum, and described on synchronized motion of two pendulum clocks hung on a common support. Even if a pendulum is artificially disturbed, they reinstated the synchronized state after a short time. ¹ Since then, the researches of synchronization have drawn the attention of an increasing number of researchers in a variety of real-world systems. In 1960s, Blekhman^2,3 proposed the synchronization theory of vibrational mechanics with two or multiple rotors—the method of direct separation of motions is the most prevalent for solving a lot of synchronous problem at that time. Subsequently, many scholars have developed other methods to explore the synchronous behavior of the rotors. Balthazar and colleagues^4,5 studied synchronization of two or four non-ideal exciters that are mounted on a flexible structural frame support by means of numerical simulations. Koluda et al.^6,7 consider the synchronization of two self-excited double-pendula hanging from the horizontally movable beam and explained how the energy is transferred between the pendula via the oscillating beam using the energy balance. Zhang et al.^8–11 employed the average method of modified small parameters to analyze coupling characteristics of two and three exciters supported on a vibrating body in a far-resonant vibrating system. In Zhang et al., ¹¹ synchronization of two exciters rotating in the same was investigated. The results show the phase difference between the two exciters in the steady-state approach to π when the swing of the rigid frame is implemented. However, in order to improve vibration amplitude of the vibrating system in engineering, the phase difference of two co-rotating exciters is expected to operate at zero phase. The smaller the absolute values of phase difference, the greater the exciting force of the system is. Hence, the model of two co-rotating exciters coupled with a spring is proposed to ensure that their phase difference is locked around zero. In recent years, the coupled rotors system is one of the study hotspots in the field of synchronous motion of rotors. Fang and colleagues^12,13 have put forward unbalanced rotors coupled with pendulum rod through a torsion spring and analyzed the synchronous characteristics in a far-resonant vibrating system by means of the Poincare method. For the research of 1-degree-of-freedom (DOF) pendulums coupled with a weak spring, speed gradient energy method is adopted to design the controller by Kumon et al.; ¹⁴ the dynamic properties of the considering whole system are investigated. In addition, Fradkov and Andrievsky ¹⁵ theoretically and numerically explored the phase relations of coupled oscillators by the phase shift between pendulum states on system parameters and initial conditions. Although the above-mentioned researches about synchronization of 1-DOF pendulums coupled with a weak spring have made great progress, the synchronization of two rotors coupled with a spring in a multi-DOF vibration system has been less reported due to the lack of comprehensive understanding about its synchronizing characteristics. In this article, we consider the model consisting of two co-rotating rotors installed with nonlinear springs in a multi-DOF vibration system and the vibrating body is connected to a fixed support by four linear springs. The average method of modified small parameters is employed to study the synchronization characteristics of the vibrating system. It is demonstrated that the value of the spring stiffness, the installed locations of two motors, and the structural parameters of the coupling unit have magnificent influence on the dynamic characteristics of the vibrating system.

The structure of the article is as follow: To begin with, we will give a description of the simplified model of two co-rotating rotors installed with nonlinear springs in a coupling vibratory system and establish dynamic equations by using Lagrange’s equations. In section “Method description,” the complex number method is employed to acquire the steady-state responses in directions of 3 degrees of freedom. In light of the average method with revised small parameter, the synchronous torque of frequency capture of the vibrating system is obtained. In addition, considering Routh–Hurwitz theorem and generalized Lyapunov equation, the synchronous criterion of the vibrating system is established. In section “Numerical results,” the results of numerical analysis in before-resonance system and after-resonance system are presented. To further verify the correctness of the above theoretical derivation and numerical computation, corresponding computer simulations are implemented in section “Computer simulation.” Finally, it is summarized in section “Conclusion.”

Model description and dynamic equations

Figure 1 shows the three-dimensional (3D) model of a vibrating system with two co-rotating rotors installed with nonlinear springs. The system consists of a vibrating body m₀ (kg), which is supported on a fixed base by means of four linear springs with stiffness coefficient k_j (N/m) and damping coefficient f_j (N s/m) in j-directions; two electric motors rotating in the same direction are parallelly installed on the vibrating body via the motor seat, which are connected with the coupling unit consisting of two nonlinear serial springs on which there are clearances d₀ (m). The simplified dynamic model of the considered system is illustrated in Figure 2, the electric motor i is represented by a mass point m_i (kg) (for i = 1, 2) and rotating radius r_i (m), whose rotational velocity is denoted by φ_i (rad/s). The coupling unit is simplified as a non-linear spring $k$ (N/m) with a clearance $d_0$ (m). The distance between the rotating center of the ith rotor and the mass center of the vibrating body is denoted by $l_i$ (m), and the installation angle of two motors is represented by ${\beta }_i$ (rad). Considering the type of motion system is the plane motion, the vibrating system can move or swing in x-, y-, and ψ-directions, respectively. Three reference frames of the system can be designed as follows: the fixed coordinate system $\mathrm{oxy}$ , the moving coordinate frame $o\prime x″y″$ , and the rotating coordinate system $o\prime x\prime y\prime$ .

Figure 1.

The 3D model of two co-rotating rotors installed with nonlinear springs.

Figure 2.

The simplified dynamic model of two co-rotating rotors installed with nonlinear springs: (a) the dynamic model and (b) three reference frames.

In quiescent state, three coordinate systems coincide with each other, and the coordinates of each motor ${\mathrm{\Phi }}_i$ and the connection ${\mathrm{\Theta }}_i$ between the ith rotor and the one end of non-linear spring are described by

$${\mathrm{\Phi }}_i={\left[\begin{array}{cc}{l}_i\cos {\beta }_i-r_i\sin {\phi }_i\hfill & l_i\sin {\beta }_i-r_i\cos {\phi }_i\hfill \end{array}\right]}^{\mathrm{T}}$$

(1)

$${\mathrm{\Theta }}_i={\left[\begin{array}{cc}{l}_i\cos {\beta }_i-a\sin {\phi }_i\hfill & l_i\sin {\beta }_i-a\cos {\phi }_i\hfill \end{array}\right]}^{\mathrm{T}}$$

(2)

Considering the displacement vector of the vibrating system is denoted by $\mathrm{\Phi }$ , during the synchronous process, the vibrating body can swing around point $o\prime$ , a new coordinate of each motor $\mathrm{\Phi }{\prime }_i$ and the connection $\mathrm{\Theta }{\prime }_i$ can be expressed by the following form through the rotating matrix K

$$\mathrm{\Phi }{\prime }_i=\mathrm{\Phi }+\mathbf{K}{\mathrm{\Phi }}_i$$

(3)

$$\mathrm{\Theta }{\prime }_i=\mathrm{\Phi }+\mathbf{K}{\mathrm{\Theta }}_i$$

(4)

where

$$\mathrm{\Phi }=\left[\begin{array}{c}x\hfill \\ \multicolumn{1}{c}{y}\end{array}\right],\mathbf{K}=\left[\begin{array}{cc}\cos \psi \hfill & \sin \psi \hfill \\ \multicolumn{1}{c}{-\sin \psi }& \cos \psi \hfill \end{array}\right]$$

(5)

According to the mechanical system dynamics basic principle, the expression for the kinetic of the system can be obtained as follows

$$T=\frac{1}{2}{m}_0{\stackrel{·}{\mathrm{\Phi }}}^{\mathrm{T}}\stackrel{·}{\mathrm{\Phi }}+\frac{1}{2}{J}_0{\stackrel{·}{\psi }}^2+\frac{1}{2}\sum _{i=1}^2{J}_{\mathrm{oi}}{\stackrel{·}{\phi }}_i^2+\frac{1}{2}{m}_i\stackrel{·}{\mathrm{\Phi }}{\prime }^{\mathrm{T}}\stackrel{·}{\mathrm{\Phi }}\prime$$

(6)

Here, $J_0$ is rotational inertia of the vibrating system and $J_{\mathrm{oi}}$ is the moments of inertia of the i-th rotor with respect to its axis, $J_{\mathrm{oi}}\approx m_i{r}_i^2$ . Considering initial distance of two rotors in x-direction, which is denoted by $\ell$ (m), the coupling spring is in a free state when two rotors are synchronously situated in the lowest position. When spring's length is less than its free length, and the spring has always kept in compressed state. On the contrary, the spring will not have any forces. The distance between the coupling unit and the rotating center of the rotor is defined by $a$ (m). Assuming $2a/\ell <<1$ , the non-linear spring is only subjected to the pressure of the load during synchronization process, and its deformation length can be obtained as follows

$$\mathrm{\Delta }=\{\begin{array}{c}\left|a(\sin {\phi }_1-\sin {\phi }_2)\right|(0\le l-\ell -0.5d_0\le l_0)\cup (0\le \ell -l-0.5d_0\le l_0)\hfill \\ \multicolumn{1}{c}{0\mathrm{other}}\end{array}$$

(7)

where

$$\begin{array}{c}\ell =\sqrt{l_1^2+l_2^2-2l_1{l}_2\cos ({\beta }_2-{\beta }_1)},\hfill \\ l=\sqrt{{({\mathrm{\Theta }\prime }_1-{\mathrm{\Theta }\prime }_2)}^{\mathrm{T}}({\mathrm{\Theta }\prime }_1-{\mathrm{\Theta }\prime }_2)}\hfill \end{array}$$

The total potential energy of the system is given by

$$V=\frac{1}{2}{k}_x{x}^2+\frac{1}{2}{k}_y{y}^2+\frac{1}{2}{k}_{\psi }{\psi }^2+\frac{1}{2}k{\mathrm{\Delta }}^2$$

(8)

And the dissipated energy of the whole system can be expressed as follows

$$D=\frac{1}{2}{f}_x{\stackrel{·}{x}}^2+\frac{1}{2}{f}_y{\stackrel{·}{y}}^2+\frac{1}{2}{f}_{\psi }{\stackrel{·}{\psi }}^2+\frac{1}{2}{f}_1{\stackrel{·}{\phi }}_1^2+\frac{1}{2}{f}_2{\stackrel{·}{\phi }}_2^2$$

(9)

In light of the general form of Lagrange’s equation

$$\frac{d}{\mathrm{dt}}\frac{\partial T}{\partial {\stackrel{·}{q}}_i}-\frac{\partial (T-V)}{\partial q_i}+\frac{\partial D}{\partial {\stackrel{·}{q}}_i}=Q_i$$

(10)

Choosing the matrix $q=[x,y,\psi ,{\phi }_1,{\phi }_2]$ as s the general coordinates and $Q=[0,0,0,M_{e1}-R_{e1},M_{e2}-R_{e2}]$ as the generalized non-conservative. In engineering, the weight of two motors is much less than that of the rigid frame, that is, $m_i<<M$ , and $\psi <<1$ , those coupling term, therefore, have been neglected. Then, taking into account relations (6), (9), (11), we arrive at the following differential equations of the vibrating system

$$\begin{array}{c}M\stackrel{··}{x}+f_x\stackrel{·}{x}+k_{x}x=\sum _{i=1}^2{m}_i{r}_i({\stackrel{··}{\phi }}_i\cos {\phi }_i-{\stackrel{·}{\phi }}_i^2\sin {\phi }_i)\hfill \\ M\stackrel{··}{y}+f_{\psi }\stackrel{·}{\psi }+k_{y}y=-\sum _{i=1}^2{m}_i{r}_i({\stackrel{··}{\phi }}_i\sin {\phi }_i+{\stackrel{·}{\phi }}_i^2\cos {\phi }_i)\hfill \\ J\stackrel{··}{\psi }+f_{\psi }\stackrel{·}{\psi }+k_{\psi }\psi =\hfill \\ \sum _{i=1}^2{m}_i{l}_i{r}_i[{\stackrel{··}{\phi }}_i\sin ({\phi }_i+{\beta }_i)+{\stackrel{·}{\phi }}_i^2\cos ({\phi }_i+{\beta }_i)]\hfill \\ J_{o1}{\stackrel{··}{\phi }}_1+f_1{\stackrel{·}{\phi }}_1=\hfill \\ M_{e1}-R_{e1}+m_1{r}_1[\stackrel{··}{x}\cos ({\phi }_1+\psi )-\stackrel{··}{y}\sin ({\phi }_1+\psi )\hfill \\ +l_1\stackrel{··}{\psi }\sin ({\phi }_1+{\beta }_1)-l_1{\stackrel{·}{\psi }}^2\cos ({\phi }_1+{\beta }_1)]\hfill \\ -k{a}^2(\sin {\phi }_1-\sin {\phi }_2)\cos {\phi }_1\hfill \\ J_{o2}{\stackrel{··}{\phi }}_2+f_2{\stackrel{·}{\phi }}_2=\hfill \\ M_{e2}-R_{e2}+m_2{r}_2[\stackrel{··}{x}\cos ({\phi }_2+\psi )-\stackrel{··}{y}\sin ({\phi }_2+\psi )\hfill \\ +\stackrel{··}{\psi }{l}_2\sin ({\phi }_2+{\beta }_2)-{\stackrel{·}{\psi }}^2{l}_2\cos ({\phi }_2+{\beta }_2)]\hfill \\ +k{a}^2(\sin {\phi }_1-\sin {\phi }_2)\cos {\phi }_2\hfill \end{array}$$

(11)

where $M$ is the whole mass of the vibrating body, $M=m_0+\sum _{i=1}^2{m}_i$ , and $J$ is the moment of inertia of whole vibrating body, $J=J_0+\sum _{i=1}^2{m}_i{l}_i^2+\sum _{i=1}^2{m}_i{r}_i^2$ , $(\stackrel{·}{•})\mathrm{and}$ $(\stackrel{··}{•})\mathrm{indicate}$ $d(•)/\mathrm{dt}$ and $d^2(•)/d{t}^2$ , respectively.

Method description

The steady approximation responses

It can be seen from equation (11) that the dynamic equations of the vibrating system are mutually coupled with respect to degree of freedom $x$ , $y$ , $\psi$ , ${\phi }_1$ , and ${\phi }_2$ . The parameters of two motors are assumed to be identical with each other, $r_1=r_2=r$ and introducing the following dimensionless parameters into the first three formulas of equation (11), we obtain

$$\begin{array}{c}{\omega }_{\mathrm{nx}}=\sqrt{\frac{k_x}{M}},{\omega }_{\mathrm{ny}}=\sqrt{\frac{k_y}{M}},{\omega }_{n\psi }=\sqrt{\frac{k_{\psi }}{J}},\hfill \\ {\varsigma }_x=\frac{f_x}{2{\omega }_{\mathrm{nx}}M},{\varsigma }_y=\frac{f_y}{2{\omega }_{\mathrm{ny}}M},{\varsigma }_{\psi }=\frac{f_{\psi }}{2{\omega }_{n\psi }J}\hfill \\ n_x=\frac{{\omega }_m}{{\omega }_{\mathrm{nx}}},n_y=\frac{{\omega }_m}{{\omega }_{\mathrm{ny}}},n_{\psi }=\frac{{\omega }_m}{{\omega }_{n\psi }},{\eta }_{12}=\frac{m_1}{m_2},\hfill \\ r_m=\frac{m_2}{M},l_e=\sqrt{\frac{J}{M}},r_{\mathrm{li}}=\frac{l_i}{l_e},(i=1,2)\hfill \end{array}$$

(12)

Since two motors synchronously operate with identical angular velocity $({\stackrel{··}{\phi }}_i<<{\stackrel{·}{\phi }}_i^2)$ , ${\stackrel{··}{\phi }}_i$ in the first three formulae of equation (11) can be ignored. Therefore, the corresponding dimensionless equations of the system in $j-(j=x,y,\psi )$ directions have the form

$$\begin{array}{c}\stackrel{··}{x}+2{\omega }_{\mathrm{nx}}{\varsigma }_x\stackrel{·}{x}+{\omega }_{\mathrm{nx}}^{2}x=-{\omega }_m^2{r}_{m}r[{\eta }_{12}\sin {\phi }_1+\sin {\phi }_2]\hfill \\ \stackrel{··}{y}+2{\omega }_{\mathrm{ny}}{\varsigma }_y\stackrel{·}{\psi }+{\omega }_{\mathrm{ny}}^{2}y=-{\omega }_m^2{r}_{m}r[{\eta }_{12}\cos {\phi }_1+\cos {\phi }_2]\hfill \\ \stackrel{··}{\psi }+2{\omega }_{n\psi }{\varsigma }_{\psi }\stackrel{·}{\psi }+{\omega }_{n\psi }^2\psi =\hfill \\ \frac{{\omega }_m^2{r}_{m}r}{l_e}[{\eta }_{12}{r}_{l1}\cos ({\phi }_1+{\beta }_1)+r_{l2}\cos ({\phi }_2+{\beta }_2)]\hfill \end{array}$$

(13)

The approximately steady response of dynamic equation (13) can be obtained by applying complex number method, and its solving process is explained in Appendix 1, for which we obtain the following expressions

$$\begin{array}{c}x=-{\mu }_x{r}_{m}r[{\eta }_{12}\sin ({\phi }_1-{\gamma }_x)+\sin ({\phi }_2-{\gamma }_x)]\hfill \\ y=-{\mu }_y{r}_{m}r[{\eta }_{12}\cos ({\phi }_1-{\gamma }_y)+\cos ({\phi }_2-{\gamma }_y)]\hfill \\ \psi =\frac{{\mu }_{\psi }{r}_{m}r}{l_e}[{\eta }_{12}{r}_{l1}\cos ({\phi }_1+{\beta }_1-{\gamma }_{\psi })\hfill \\ +r_{l2}\cos ({\phi }_2+{\beta }_2-{\gamma }_{\psi })]\hfill \end{array}$$

(14)

where

$$\begin{array}{c}{\mu }_j=\frac{n_j^2}{\sqrt{{(1-n_j^2)}^2+{\left(2{\varsigma }_j{n}_j\right)}^2}},\hfill \\ {\gamma }_j=\{\begin{array}{c}\arctan \left(\frac{2{\varsigma }_j{n}_j}{1-n_j^2}\right)1-n_j^2\ge 0\hfill \\ \multicolumn{1}{c}{\pi +\arctan \left(\frac{2{\varsigma }_j{n}_j}{1-n_j^2}\right)1-n_j^2<0}\end{array}(j=x,y,\psi )\hfill \end{array}$$

Synchronization condition

Assuming the average phase of two motors and their phase difference to be $\phi$ and $2\alpha$ , respectively, the angular displacements of two motors can be presented in the following form

$$\begin{array}{c}{\phi }_1=\phi +\alpha \hfill \\ {\phi }_2=\phi -\alpha \hfill \end{array}$$

(15)

In light of the literature, ² it should be noted that ${\stackrel{·}{\phi }}_i$ with respect to a periodic function of time is a slowly varying parameter. Due to the periodic motion of the vibrating system, the angular velocity of two motors also changes periodically. Assigning the least positive period of two motors as $T_{\mathrm{LCMP}}$ , in which their average angular velocity over a period of time are approximately equal to a constant, we have

$${\omega }_m=\frac{1}{T_{\mathrm{LCMP}}}\underset{t}{\overset{t+T_{\mathrm{LCMP}}}{\int }}\phi \left(t\right)\mathrm{dt}=\mathrm{costant}$$

(16)

Considering the instantaneous change coefficients of $\stackrel{·}{\phi }$ and $\stackrel{·}{\alpha }$ expressed as ${\epsilon }_1$ and ${\epsilon }_2$ ( ${\epsilon }_1$ and ${\epsilon }_2$ are small parameters with respect to time $t$ ), that is, $\stackrel{·}{\phi }=(1+{\epsilon }_1){\omega }_m$ , $\stackrel{·}{\alpha }={\epsilon }_2{\omega }_m$ , we introduce two new parameters ${\nu }_1$ and ${\nu }_2$ , that is

$$\begin{array}{c}{\nu }_1={\epsilon }_1+{\epsilon }_2\hfill \\ {\nu }_2={\epsilon }_1-{\epsilon }_2\hfill \end{array}$$

(17)

From equations (15) and (17), we obtain the following rotating acceleration of two motors

$$\begin{array}{c}{\stackrel{·}{\phi }}_1=(1+{\nu }_1){\omega }_m\hfill \\ {\stackrel{·}{\phi }}_2=(1+{\nu }_2){\omega }_m\hfill \end{array}$$

(18)

When the vibrating system operates at the steady state, the average value of the instantaneous change coefficients of two motors over one period $(T_0=2\pi /{\omega }_m)$ is equal to zero, that is, ${\overline{\epsilon }}_i=(1/T){\int }_0^{T_0}{\epsilon }_i\mathrm{dt}$ $(i=1,2)$ . In addition, while two motors rotate with the average angular velocities $({\omega }_m)$ in the steady state, those parameters ${\epsilon }_i$ , ${\stackrel{·}{\epsilon }}_i$ , ${\nu }_i$ , ${\overline{\nu }}_i$ , and $\overline{\alpha }$ are small compared with the changing ${\phi }_i$ . Thus, we regard them as slow-changing parameters. And the middle values of their integrations over a single period are expressed as ${\overline{\epsilon }}_i$ , $\stackrel{·}{\overline{{\epsilon }_i}}$ , ${\overline{\nu }}_i$ , $\stackrel{·}{\overline{{\nu }_i}}$ , and $\overline{\alpha }$ , respectively. Taking into account equation (14) by the chain rule, we obtain $\stackrel{··}{x}$ , $\stackrel{··}{y}$ , and $\stackrel{··}{\psi }$ . Substituting them into the dynamic equations of two motors in equation (11), which are integrated over one period $(T_0=2\pi /{\omega }_m)$ , the balanced equations of two motors are transformed to the form

$$\begin{array}{c}{J}_{o1}{\omega }_m\stackrel{·}{\overline{{\nu }_1}}+f_1{\omega }_m(1+{\overline{\nu }}_1)+{\overline{T}}_{L1}=T_{e1}\hfill \\ J_{o2}{\omega }_m\stackrel{·}{\overline{{\nu }_2}}+f_1{\omega }_m(1+{\overline{\nu }}_2)+{\overline{T}}_{L2}=T_{e2}\hfill \end{array}$$

(19)

with

$$\begin{array}{c}{\overline{T}}_{L1}={\lambda }_{11}\prime \stackrel{·}{\overline{{\nu }_1}}+{\lambda }_{12}\prime \stackrel{·}{\overline{{\nu }_2}}+{\lambda }_{13}\prime {\overline{\nu }}_1+{\lambda }_{14}\prime {\overline{\nu }}_2+{\lambda }_{15}\prime \hfill \\ {\overline{T}}_{L2}={\lambda \prime }_{21}\stackrel{·}{\overline{{\nu }_1}}+{\lambda }_{22}\prime \stackrel{·}{\overline{{\nu }_2}}+{\lambda }_{23}\prime {\overline{\nu }}_1+{\lambda }_{24}\prime {\overline{\nu }}_2+{\lambda }_{25}\prime \hfill \end{array}$$

(20)

where those coefficients in the right-hand side of equation (20) are listed in Appendix 2. We note that $\lambda {\prime }_{\mathrm{ik}}(i=1,2;k=1,2,3,4,5)$ are influenced by $\overline{\alpha }$ , $W_{\mathrm{ci}}$ , $W_{\mathrm{si}}$ , $a_s$ , $b_s$ , $a_c$ , and $b_c$ . However, $W_{\mathrm{si}}$ represents the sine coefficient of the parameters ${\gamma }_j(j=x,y,\psi )$ , and $W_{\mathrm{ci}}$ are regarded as its cosine coefficient. As $W_{\mathrm{si}}$ is much small compared with $W_{\mathrm{ci}}$ , the terms related to the sine coefficient can be neglected. The efficiently torque of two motors in equation (19) can be linearized in the vicinity ${\stackrel{·}{\phi }}_i\approx {\omega }_m$ , that is

$$\begin{array}{c}{T}_{e1}=T_{e01}-k_{e1}({\overline{\epsilon }}_1+{\overline{\epsilon }}_2)=T_{e01}-k_{e1}{\overline{\nu }}_1\hfill \\ T_{e2}=T_{e02}-k_{e2}({\overline{\epsilon }}_1-{\overline{\epsilon }}_2)=T_{e02}-k_{e2}{\overline{\nu }}_2\hfill \end{array}$$

(21)

where $T_{e0i}$ is the driving torque of the ith motor, and $k_{\mathrm{ei}}$ is the stiffness coefficient of angular velocity when the two motors are consistently rotate with a synchronous speeds ${\omega }_m$ . Now employing formulas (21) and (19), we get the frequency capture equation of the two motors with the matrix form

$$\mathbf{A}\stackrel{·}{\mathbf{\nu }}=\mathbf{B}\mathbf{\nu }+\mathbf{\mu }$$

(22)

where

$$\begin{array}{c}\mathbf{v}={[{\overline{\nu }}_1,{\overline{\nu }}_2]}^{\mathrm{T}},\mathbf{\mu }={[{\mu }_1,{\mu }_2]}^{\mathrm{T}},\hfill \\ \mathbf{A}=\left[\begin{array}{cc}{a}_{11}\hfill & a_{12}\hfill \\ \multicolumn{1}{c}{a_{21}}& a_{22}\hfill \end{array}\right],\mathbf{B}=\left[\begin{array}{cc}{b}_{11}\hfill & b_{12}\hfill \\ \multicolumn{1}{c}{b_{21}}& b_{22}\hfill \end{array}\right]\hfill \end{array}$$

with

$$\begin{array}{c}{\rho }_1={\eta }_{12}+\frac{1}{2}{\eta }_{12}^2{W}_{c1};{\rho }_2=1+\frac{1}{2}{W}_{c2}\hfill \\ k_1={\eta }_{12}^2{W}_{s1}+\frac{k_{e1}}{m_2{r}^2{\omega }_m^2}+\frac{f_1}{m_2{r}^2{\omega }_m};\hfill \\ k_2=W_{s2}+\frac{k_{e2}}{m_2{r}^2{\omega }_m^2}+\frac{f_2}{m_2{r}^2{\omega }_m}\hfill \\ {\mu }_1=\frac{T_{e01}}{m_2{r}^2{\omega }_m}-\frac{f_1}{m_2{r}^2}-\frac{1}{2}{\eta }_{12}^2{\omega }_m{W}_{s1}+\frac{T_{e02}}{m_2{r}^2{\omega }_m}\hfill \\ -\frac{f_2}{m_2{r}^2}-\frac{1}{2}{\omega }_m{W}_{s2}-{\omega }_m{W}_c\cos (2\alpha +{\theta }_c)\hfill \\ {\mu }_2=\frac{T_{e01}}{m_2{r}^2{\omega }_m}-\frac{f_1}{m_2{r}^2}-\frac{1}{2}{\eta }_{12}^2{\omega }_m{W}_{s1}-\frac{T_{e02}}{m_2{r}^2{\omega }_m}+\frac{f_2}{m_2{r}^2}\hfill \\ +\frac{1}{2}{\omega }_m{W}_{s2}-{\omega }_m{W}_s\sin (2\alpha -{\theta }_s)-\frac{k{a}^2\sin \left(2\alpha \right)}{m_2{r}^2{\omega }_m}\hfill \end{array}$$

In equation (22), symbol A represents inertial coupling matrix of two motors. Symbol B is defined as stiffness matrix of two motors. And symbol $\mathbf{\mu }$ is matrix with respect to the driving torque and the resistance torque of the two motors. If the vibrating system operating in the steady state, we have ${\overline{\epsilon }}_1=0$ and ${\overline{\epsilon }}_2=0$ , thereby, $\mathbf{\mu }=0$ . Substituting it into equation (22), we obtain the following equations

$$\begin{array}{c}{T}_{e01}+T_{e02}-(f_1+f_2){\omega }_m-\frac{1}{2}{m}_2{r}^2{\omega }_m^2[{\eta }_{12}^2{W}_{s1}+W_{s2}]\hfill \\ -m_2{r}^2{\omega }_m^2{W}_c\cos (2\alpha +{\theta }_c)=0\hfill \\ (T_{e01}-T_{e02})-{\omega }_m(f_1-f_2)+\frac{1}{2}{m}_2{r}^2{\omega }_m^2(W_{s2}-{\eta }_{12}^2{W}_{s1})\hfill \\ =m_2{r}^2{\omega }_m^2{W}_s\sin (2\alpha -{\theta }_s)+k{a}^2\sin \left(2\alpha \right)\hfill \end{array}$$

(23)

As seen from equation (23), the fist formula is the equilibrium equation for the inertial moment of the vibrating system in the steady state. The first term $T_{e01}+T_{e02}$ represents the sum of the driving torque of two motors and the second term $(f_1+f_2){\omega }_m$ is the sum of the resisting moment of two motors. The last term is the sum of the load torques that vibrating system acts on two motors. It also can be seen that the residual torques are balanced by adjusting the load torques of the system. In addition, the second formula is the differential equation between motors 1 and 2. Rewriting it, we have

$$\frac{T_D}{T_c}=\sin (2\alpha -{\theta }_M)$$

(24)

where

$$\begin{array}{c}{T}_c=\sqrt{{(m_2{r}^2{\omega }_m^2{W}_s\cos {\theta }_s+k{a}^2)}^2+{(m_2{r}^2{\omega }_m^2{W}_s\sin {\theta }_s)}^2},\hfill \\ T_D=T_{R1}-T_{R2},\hfill \\ T_{R1}=T_{e01}-{\omega }_m{f}_1-\frac{1}{2}{m}_2{r}^2{\omega }_m^2{\eta }_{12}^2{W}_{s1},\hfill \\ T_{R2}=T_{e02}-{\omega }_m{f}_2-\frac{1}{2}{m}_2{r}^2{\omega }_m^2{W}_{s2}\hfill \\ {\theta }_M=\{\begin{array}{c}\arctan \frac{m_2{r}^2{\omega }_m^2{W}_s\sin {\theta }_s}{m_2{r}^2{\omega }_m^2{W}_s\cos {\theta }_s+k{a}^2},\hfill \\ \multicolumn{1}{c}{m_2{r}^2{\omega }_m^2{W}_s\cos {\theta }_s+k{a}^2\ge 0}\\ \multicolumn{1}{c}{\pi +\arctan \frac{m_2{r}^2{\omega }_m^2{W}_s\sin {\theta }_s}{m_2{r}^2{\omega }_m^2{W}_s\cos {\theta }_s+k{a}^2},}\\ \multicolumn{1}{c}{m_2{r}^2{\omega }_m^2{W}_s\cos {\theta }_s+k{a}^2<0}\end{array}\hfill \end{array}$$

Here, $T_c$ is the synchronous torque of frequency capture of the vibrating system, $T_D$ is the difference of the residual torque of two motors, and $T_{R1}$ and $T_{R2}$ are the residual torque of motors 1 and 2, respectively. As you can see, the phase difference $2\alpha$ is decided by the parameters $T_D$ , $T_c$ , and ${\theta }_M$ , which are the functions of parameters $m_i$ , $r$ , $k$ , $l_i$ , and ${\beta }_i$ . Since $|\sin (2\alpha -{\theta }_M)|\le 1$ , the synchronous condition of the system can be written in the following form

$$\left|T_D\right|\le T_c$$

(25)

From equation (25), in order to guarantee the synchronous operation between the rotors in the steady state, the synchronous torque of two motors must be greater than or equal to the difference of their residual torque.

Stability of synchronous state

Linearizing equation (22) in the vicinity of $\overline{\alpha }={\overline{\alpha }}_0$ by using the Taylor series expansion and attaching $\mathrm{\Delta }\stackrel{·}{\alpha }={{\omega }^{*}}_m{\overline{\epsilon }}_2$ $(\mathrm{\Delta }\alpha =\overline{\alpha }-{\overline{\alpha }}_0)$ as the third row, three linear differential equations of the first order of two motors can be obtained

$$\mathbf{A}\mathbf{\text{'}}\stackrel{·}{\mathbf{z}}=\mathbf{B}\mathbf{\text{'}}\mathbf{z}$$

(26)

where

$$\begin{array}{c}\mathbf{z}=[{\overline{\epsilon }}_1,{\overline{\epsilon }}_2,\overline{\alpha }-{\overline{\alpha }}_0],\mathbf{A}\mathbf{\text{'}}=\left[\begin{array}{ccc}{a\prime }_{11}\hfill & a_{12}\prime \hfill & 0\hfill \\ \multicolumn{1}{c}{{a\prime }_{21}}& a_{22}\prime \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right],\hfill \\ \mathbf{B}\mathbf{\text{'}}=\left[\begin{array}{ccc}{b\prime }_{11}\hfill & b_{12}\prime \hfill & b_{13}\prime \hfill \\ \multicolumn{1}{c}{{b\prime }_{21}}& b_{22}\prime \hfill & b_{23}\prime \hfill \\ \multicolumn{1}{c}{0}& {\omega }_m^{*}\hfill & 0\hfill \end{array}\right]\hfill \end{array}$$

The parameters of the matrices $\mathbf{A}\mathbf{\text{'}}$ and $\mathbf{B}\mathbf{\text{'}}$ can be found in Appendix 4. Assuming the exponential time-dependence has the form $z=u\exp (\lambda t)$ and solving the characteristic equation $|{\mathbf{A}\mathbf{\text{'}}}^{-1}\mathbf{B}\mathbf{\text{'}}-\lambda \mathbf{E}|=0$ , we get the following expression for characteristic equation

$${\lambda }^3+c_1{\lambda }^2+c_2\lambda +c_3=0$$

(27)

where

$$\begin{array}{c}{c}_1=\frac{4{\omega }_m{H}_1}{H_0},c_2=\frac{2{\omega }_m^2{H}_2}{H_0},c_3=\frac{2{\omega }_m^3{H}_3}{H_0}\hfill \\ H_0=4{\rho }_1{\rho }_2-W_s^2{\cos }^2(2\alpha -{\theta }_s)+W_c^2{\sin }^2(2\alpha +{\theta }_c)\hfill \\ H_1={\rho }_2{k}_1+{\rho }_1{k}_2-W_s{W}_c\cos ({\theta }_c+{\theta }_s)\hfill \\ H_2=2k_1{k}_2+W_s^2{\sin }^2(2\alpha -{\theta }_s)-W_c^2{\cos }^2(2\alpha +{\theta }_c)\hfill \\ +W_s^2-W_c^2+({\rho }_1-{\rho }_2)W_c\sin (2\alpha +{\theta }_c)\hfill \\ +({\rho }_1+{\rho }_2)W_s\cos (2\alpha -{\theta }_s)\hfill \\ +[({\rho }_1+{\rho }_2)+W_s\cos (2\alpha -{\theta }_s)]\frac{k{a}^2}{m_2{r}^2{\omega }_m^2}\cos \left(2\alpha \right)\hfill \\ H_3=[k_1+k_2]W_s\cos (2\alpha -{\theta }_s)\hfill \\ -[k_2-k_1]W_c\sin (2\alpha +{\theta }_c)+2W_c{W}_s\cos ({\theta }_s+{\theta }_c)\hfill \\ +[k_1+k_2+2W_c\cos (2\alpha +{\theta }_c)]\frac{k{a}^2}{m_2{r}^2{\omega }_m^2}\cos \left(2\alpha \right)\hfill \end{array}$$

If the real parts of all roots are negative, the zero solution of equation (26) is stable, that is, $\mathbf{z}=0$ , asymptotically stable periodic solution equation (11) exists. If there is at least one root of equation (27) whose real part is positive, the corresponding solution is unstable. According to the Routh–Hurwitz criterion, the synchronous criterion of the vibrating system can be written as follows

$$c_1>0,c_3>0,c_1{c}_2>c_3$$

(28)

So, we obtain condition of self-synchronization motion stability in the following form

$$\begin{array}{c}{H}_0>0,H_1>0,H_3>0,4H_1{H}_2>H_0{H}_3\hfill \\ H_0<0,H_1<0,H_3<0,4H_1{H}_2>H_0{H}_3\hfill \end{array}$$

(29)

Analyzing equations (24) and (29), it is indicated that the synchronous state and the stability criterion of the vibrating system are influenced by the frequency ratios $n_j(j=x,y,\psi )$ , the positional parameters of two motors $(r_{\mathrm{li}},{\beta }_i)$ , and the structural parameters of the coupling unit $(d_0,l_0)$ .

Numerical results

Some theoretical derivation in simplified form on synchronization problem for the non-resonance system is given in the preceding sections. In order to clearly master the influence of some parameters of the vibrating system on the synchronous behavior, in this section, the quantitative discussion on solving the phase difference of two motor by numerical method is presented. In light of Zhao et al., ¹⁶ while two motors rotate with the same absolute value of their average angular velocities, the electromagnetic torque and the stiffness coefficients in the steady state can be simplified as follows

$$T_{\mathrm{ek}}=n_p\frac{L_{\mathrm{mk}}{U}_{\mathrm{so}}^2}{L_{\mathrm{sk}}{\omega }_s{R}_{\mathrm{rk}}}({\omega }_s-n_p{\omega }_m)$$

(30)

$$k_{\mathrm{ek}}={n_p}^2\frac{L_{\mathrm{mk}}^2{U}_{\mathrm{so}}^2}{L_{\mathrm{sk}}{\omega }_s{R}_{\mathrm{rk}}}$$

(31)

where $U_{\mathrm{so}}$ is the amplitude of the stator voltage vector, $L_{\mathrm{mk}}$ is the mutual inductance of the kth motor, $n_p$ is the number of pole-pairs of the motors, $R_{\mathrm{rk}}$ is the rotor resistance of the kth motor, ${\omega }_s$ is the synchronous electric angular velocity, $L_{\mathrm{sk}}$ is the stator inductance of the kth motor. In this section, we consider the model shown in Figure 1 and assume the parameters of two motors are identical, ${\omega }_x=50$  Hz, $U_{\mathrm{so}}=220$  V, $n_p=2$ , $R_{\mathrm{si}}=0.54$ , $L_m=0.13$  H, $L_r=0.2$ , and $L_{\mathrm{sk}}=0.1$  H. Other parameters of the vibrating system are given in Table 1. According to equation (12), the dimensionless parameters corresponding to system parameters are listed in Table 2.

Table 1.

Values for system parameters of the nonlinear vibrating system.

Unbalanced rotors $i=1,2$	Vibro-mass	Damping spring	Installation parameters	Coupling unit
$m_i=2$  kg	$M=70$  kg	$k_x=1.7\times {10}^4~8\times {10}^7$  N/m	$l_1=0.42,1.09$	$k=1~1\times {10}^8$  N/m
$r=0.04$  m	$J=6.8$  kg m 2	$k_y=1.7\times {10}^4~8\times {10}^7$  N/m	$1.52,2.12$  m	$a=0.02$  m
${\omega }_m=157$  rad/s	–	$k_{\psi }=1.6\times {10}^3~1.7\times {10}^7$  N/rad	$l_2=0.42,1.09$	$d_0=0~0.08$  m
$c_i=0.02$  N s/m	–	$f_x=1000$  N s/m	$1.52,2.12$  m	$l_0=0.04~0.07$  m
–	–	$f_y=1000$  N s/m	${\beta }_1=65°,30°$
–	–	$f_{\psi }=1000$  N s/rad	${\beta }_2=115°,150°$

Table 2.

Values for dimensionless parameters according to equation (12).

${\eta }_{12}=1$

$r_m=0.028$

$l_e=0.31$

$r_l=0.3-7$

$n_x=0.1~10$

$n_y=0.1~10$

$n_{\psi }=0.1~10$

Numeric analysis in a nonlinear system

For the vibrating system in Figure 1, the coupling unit that its structural parameters satisfy $2l_0+d_0=0.14$  m is made up of in series with two springs, and there exists a clearance $d_0$ (m) on between the two springs. As seen from equation (7), during the running process of the steady state of the vibrating system, the deformation length of the coupling spring is not always equal to a constant, which are influenced by the structural parameters of the coupling unit and the location parameters of two motors. Taking into account the system parameters in Table 1, when corresponding structural parameters of the coupling unit makes the vibrating system operating at the synchronous state, we can solve out $d_0\ge 1.42$  m or $d_0\le 2\times {10}^{-5}$  m. As the mechanical model in Figure 1 satisfies $d_0\le 0.14$  m, so $d_0\le 2\times {10}^{-5}$  m. The results show that if the clearance between in series with two springs exists, the value of $d_0$ should be little to ensure vibrating system operates at the steady state. Therefore, in this case, we assume $d_0=0$ . In addition, according to the range of frequency ration, the coupling type of the system can be defined as three types: before-resonance system $(0.1\le n_j\le 0.9)$ , after-resonance system $(1.1\le n_j)$ , and resonance system $(0.9\le n_j\le 1.1)$ . In following sections, we will quantitatively analyze the influence of the sable phase difference on system parameters in a non-resonance system. Restricting $0\le n_x=n_y=n_{\psi }\le 10$ , $0\le k\le 8\times {10}^5$  N/m and $r_{\mathrm{li}}$ (let $r_{\mathrm{li}}$ be 1.34, 3.5, 4.8, 6.8, respectively). The cloud picture about the stable phase difference between two motors is determined by the frequency ration, the stiffness coefficient of the coupled nonlinear spring and the location parameters of two motors are depicted in Figures 3 and 4.

Figure 3.

The stable phase difference with the theoretical computation for ${\beta }_1=150°,{\beta }_2=30°$ : (a) $r_{l1}=r_{l2}=1.34$ , (b) $r_{l1}=r_{l2}=3.5$ , (c) $r_{l1}=r_{l2}=4.8$ , and (d) $r_{l1}=r_{l2}=6.8$ .

Figure 4.

Stable phase difference with the theoretical computation for ${\beta }_1=115°,{\beta }_2=65°$ : (a) $r_{l1}=r_{l2}=1.34$ , (b) $r_{l1}=r_{l2}=3.5$ , (c) $r_{l1}=r_{l2}=4.8$ , and (d) $r_{l1}=r_{l2}=6.8$ .

From Figure 3, here ${\beta }_1=150°,{\beta }_2=30°$ , it can be seen that the stable phase difference satisfies $2\alpha \in (-\pi ,0)$ in before-resonance system, and $2\alpha$ tends to zero with the increasing in the stiffness of coupling spring. However, the stable phase difference between the two motors satisfies $2\alpha \in (0,\pi )$ in after-resonance system, $2\alpha$ is also close to zero when the stiffness of coupling spring increases gradually. In addition, comparing the effect of Figure 3, it is demonstrated that the parameter $r_l$ has influence on the stable phase difference. Under the condition of $k=0$ , the value of $r_l$ is small, the motion of the vibrating system trends to antiphase synchronization ( $|2\alpha |\in (\pi /2,3\pi /2)$ ), that is to say, the closer the distance between two motors, the stable phase difference approximates to zero.

On the other hand, from Figure 4, it follows the influence of the parameters $n_j(j=x,y,\psi )$ , $r_{\mathrm{li}}(i=1,2)$ , $k$ on the value of the phase difference for ${\beta }_1=115°,{\beta }_2=65°$ . The similar results in Figure 3 are obtained. Moreover, from Figure 4(a) when $r_{l1}=r_{l2}=1.34$ (i.e. $l_1=l_2=0.42$  m), the in-phase synchronization ( $|2\alpha |\in (0,\pi /2)$ ) occurs in before-resonance system. But the anti-phase synchronization occurs in after-resonance system for “small” coupling $k$ (i.e. $k=0$  N/m) and the synchronous state is suddenly jumped to the in-phase synchronization with the decreasing $k$ . Compared with the results in Figure 3, it can be found that the parameters ${\beta }_i(i=1,2)$ has also influence on the synchronous state in the steady operation of the system. In order to further understand the effectiveness of system parameter on stable phase difference in the steady state, the numerical discussions in before-resonance system and after-resonance system are presented in the following section.

Numerical discussions in before-resonance system

In this section, we explain the influence of the parameters $\eta$ , $r_l$ , ${\beta }_1$ , ${\beta }_2$ , and $k$ on the synchronizing characteristics of the system in detail. Since the frequency ration $n_j(j=x,y,\psi )$ in before-resonance system satisfies $0.1<n_j<0.9$ , we consider $n_j$ is equal to 0.2, 0.3, 0.5, 0.6, respectively. Here, restricting ${\beta }_1=150°,{\beta }_2=30°$ , the graphs about the approximate theoretical value of the phase difference versus the stiffness of the coupling spring can be depicted in Figures 5 and 6. From Figure 5(a), for $n_x=n_y=n_{\psi }=0.2$ , if $r_{l1}=r_{l2}=1.34$ (i.e. $l_1=l_2=0.42$  m) and $k=0$ , the phase difference $2\alpha$ approximately stabilizes at −1 rad Moreover, $2\alpha$ gradually increases from −1 rad to zero with the increase in the value of $k$ ; if $r_{l1},r_{l2}$ are equal to 3.5, 4.5, 6.8, respectively, and $k=0$ , the phase difference $2\alpha$ is always close to −2 rad and the phase difference simultaneously approximately stabilized at 0 rad with the increase in the value of $k$ . By analyzing the statement given above in Figure 5, we may draw conclusion that the parameters $n_x,n_y,n_{\psi }$ have little influence on the stable phase difference between the two motors. As $r_{l1},r_{l2}$ are functions of parameters $l_1,l_2$ , and $l_e$ , the phase difference is determined by the mass of the vibrating body and the positional parameters of two motors (i.e. parameters $l_1$ and $l_2$ ).

Figure 5.

The approximate theoretical value of the phase difference versus the stiffness of $k$ for ${\beta }_1=150°,{\beta }_2=30°$ : (a) $n_x=n_y=n_{\psi }=0.2$ , (b) $n_x=n_y=n_{\psi }=0.3$ , (c) $n_x=n_y=n_{\psi }=0.5$ , and (d) $n_x=n_y=n_{\psi }=0.6$ .

Figure 6.

The approximate theoretical value of the phase difference versus the stiffness of $k$ for ${\beta }_1=115°,{\beta }_2=65°$ : (a) $n_x=n_y=n_{\psi }=0.2$ , (b) $n_x=n_y=n_{\psi }=0.3$ , (c) $n_x=n_y=n_{\psi }=0.5$ , and (d) $n_x=n_y=n_{\psi }=0.6$ .

Figure 6 describes the approximate theoretical value of the phase difference $2\alpha$ versus the stiffness of $k$ for ${\beta }_1=115°,{\beta }_2=65°$ . The similar conclusions in Figure 5 can be obtained. Besides, comparing with the numerical results in Figure 5, it can be found that the parameters ${\beta }_1$ and ${\beta }_2$ also have effects on the phase difference of two motors in the vibrating system.

Numerical discussions in after-resonance system

In after-resonance system, the frequency ration $n_j$ satisfies $n_j>1.1$ , we here fix that $n_x,n_y,n_{\psi }$ are equal to 2, 3, 5, 6, respectively. Figure 7, restricting ${\beta }_1=150°,{\beta }_2=30°$ , explored the influence of parameters $n_x,n_y,n_{\psi }$ , $r_{l1},r_{l2}$ , and $k$ on stable phase difference of two motors. It can be found that the phase difference is near to $\pi$  rad when $r_{l1}=r_{l2}=1.34$ and $k=0$ , $2\alpha$ gradually decreases from $\pi$  rad to zero with the increase in the value of $k$ ; if $r_{l1},r_{l2}$ are equal to 3.5, 4.8, 6.8, respectively, the phase difference $2\alpha$ approximately stabilizes at 1.5 rad when $k=0$ . In short, the value of the phase difference $2\alpha$ belongs to the interval of $(0,\pi )$ in after-resonance system, which gradually tends to zero with the increase in $k$ . In addition, Figure 8, considering ${\beta }_1=115°,{\beta }_2=65°$ , shows the plots of the approximate theoretical value of the phase difference versus the stiffness of the coupling spring $k$ for the different frequency ration and positional parameters. Comparing with the results in Figure 7, it indicates that the parameters ${\beta }_1$ and ${\beta }_2$ weakly affect the stable phase difference in after-resonance system.

Figure 7.

The approximate theoretical value of the phase difference versus the stiffness of $k$ for ${\beta }_1=150°,{\beta }_2=30°$ : (a) $n_x=n_y=n_{\psi }=2$ , (b) $n_x=n_y=n_{\psi }=3$ , (c) $n_x=n_y=n_{\psi }=5$ , and (d) $n_x=n_y=n_{\psi }=6$ .

Figure 8.

The approximate theoretical value of the phase difference versus the stiffness of $k$ for ${\beta }_1={115}^{°},{\beta }_2={65}^{°}$ : (a) $n_x=n_y=n_{\psi }=2$ , (b) $n_x=n_y=n_{\psi }=3$ , (c) $n_x=n_y=n_{\psi }=5$ , and (d) $n_x=n_y=n_{\psi }=6$ .

Computer simulation

To further verify the correctness of the above theoretical analysis, in this section, we will set up an electromechanical coupling model whose rotors are driven by two asynchronous motors with same parameters, using the Runge–Kutaa routine to obtain the stable solution of differential equations (11). Here, the system parameters are listed in Tables 1 and 2.

Simulation results in before-resonance system

Simulation results for $n_x=n_y=n_{\psi }=0.496$ , $r_{l1}=r_{l2}=4.8$ , ${\beta }_1=150°,{\beta }_2=30°$ , $k=2.4\times {10}^5$  N/m are shown in Figure 9; here, $k_x=k_y=7.01\times {10}^6$  N/m, $k_{\psi }=6.77\times {10}^{5}N/m$ , $l_1=l_2=$ 1.52 m, and other related parameters are listed in Table 1. When two induction motors are simultaneously supplied with the electromagnetic force, the angular accelerations of the two motors are identical to each other due to the vibrating parameters of two induction motors that have identical configurations. During the staring process of the system, the driving torque of two motors acting on the massive vibrating body is very small, the velocity difference exists between the two motors. At 4 s, as shown in Figure 9(b), the synchronous behavior of the system is implemented and their synchronous velocity is observed to be nearly 152 rad/s. In addition, during the synchronous process of the system, the electromagnetic torque of motor 1 and motor 2, maintaining the vibrating system operates at the steady state, are gradually stabilized at 4.9 and 4.5 N m], respectively, as illustrated in Figure 9(a). Meanwhile, it follows from Figure 9(c) that the value of stable phase difference between the two motors stabilized in the vicinity −0.12 rad, which is in agreement with the solution that can be derived by numerical calculation (the approximate values of the theoretical analysis are near to −0.1485 rad in Figure 3(c)). As the stiffness value of supporting springs is very large in before-resonance system, the amplitudes of the steady-state response of the vibrating body are very small; it follows from Figure 9(e) that the amplitude of the vibrating system in the x-, y-, and ψ-directions is 6.8 × 10⁻⁴ m, 6.9 × 10⁻⁴ m, and 5.7 × 10⁻³ rad, respectively, and the trajectory moving of the system is elliptical motion.

Figure 9.

Simulation results in before-resonance system: (a) electromagnetic torques of the two asynchronous motors, (b) rotational velocities of the two rotors, (c) phase difference between the two rotors, (d) stable motion of vibration system in $\mathrm{xoy}$ plane, and (e) response phase portrait of vibrating body in $-j(j=x,y,\psi )$ directions.

Simulation results in after-resonance system

Figure 9 shows the simulation results for $n_x=n_y=n_{\psi }=4.456$ , $r_{l1}=r_{l2}=1.34$ , $k=8\times {10}^4$  N/m, ${\beta }_1=115°,{\beta }_2=65°$ in after-resonance system. For this case, $k_x=k_y=8.7\times {10}^4$  N/m, $k_{\psi }=8.39\times {10}^3$  N/m, $r_{l1}=r_{l2}=0.42\mathrm{m}$ , and other parameters are shown in Table 1. In the initial process of the vibrating system, owing to the total mass of the system acting on the two motors is very large, the two motors in the same direction cannot rotate with an identical velocity. After about 3 s, the synchronous phenomena of the system occurs, and the rotational velocity of two induction motors approaches to 150 rad/s, as illustrated in Figure 10(a); the output torque keeping the value of phase difference to stabilize at 0.07 rad is approximately 5.1 N m in Figure 10(b). In this case, when vibrating system operates synchronously in the steady state, the oscillating body moves in a stable elliptical track, and the amplitude of the vibrating system in the x-, y-, and ψ-directions are 2.5 × 10⁻³ m, 2.41 × 10⁻³ m, 6.8 × 10⁻³ rad, respectively. Comparing with the numerical results in Figure 4(a), under the same condition, the theoretical value of $2\alpha$ corresponding to the computer simulations is approximately equal to 0.06377 rad. In addition, the fact demonstrates the value of responsive amplitude of the vibrating system in after-resonance system is larger than that in before-resonance system.

Figure 10.

Simulation results in after-resonance system: (a) electromagnetic torques of the two asynchronous motors, (b) rotational velocities of the two rotors, (c) phase difference between the two rotors, (d) stable motion of vibration system in $\mathrm{xoy}$ plane, and (e) response phase portrait of vibrating body in $-j(j=x,y,\psi )$ directions.

Conclusion

In this article, we theoretically and numerically analyze the synchronicity of two co-rotating rotors installed with nonlinear springs in a non-resonance system by using the average method of modified small parameters, and the following conclusions can be stressed.

In order to avoid anti-phase synchronization for two co-rotating rotors system that occurs in engineering, we proposed a mechanical model of two co-rotating rotors installed with nonlinear springs and deduced the non-dimensional coupling equations of two motors in considering case, on which synchronization problem is converted into that of existence and stability of zero solution for the non-dimensional coupling equations of angular velocity. It can be seen that the synchronous torque of two motors must be greater than or equal to the difference of their residual torque when synchronous behavior of the system is implemented. The research results show that the coupling unit, coupling coefficients, and the positional parameters of two rotors have influence on the dynamic characteristics of the vibrating system, and there are clearances in between two nonlinear serial springs, which results in synchronization of the vibrating system that lies in an uncertain state. It also can be found that the stable phase difference between the two motors satisfies $2\alpha \in (-\pi ,0)$ in before-resonance system and the stable phase difference satisfies $2\alpha \in (0,\pi )$ in after-resonance system, and $2\alpha$ tends to zero with the increase in the stiffness of coupling spring. In other words, the proposed mechanism by adjusting the value of the coupling spring stiffness can make two motors operate at zero phase differences to meet the requirements of the strongly exciting force in engineering. Finally, computer simulation results are employed to verify the theoretical conclusions.