Research on synchronization and steady-state responses of a two-body vibration system driven by two co-rotating eccentric rotors mounted on different bodies

Abstract

In this paper, a new two-body self-synchronization vibration system characterized by two ideal vibration-exciting motors (asynchronous motors) installed on different vibrating bodies is investigated. Firstly, differential equations of the system are established via the Lagrange equation. Moreover, natural frequencies and stable solutions are obtained. Then, the conditions for the system to implement stable self-synchronization are derived using the equilibrium point theory of the dynamics system and Lyapunov’s stability theory. The effects of spring stiffness on the natural frequencies and synchronous motion are numerically analyzed. The steady-state responses of the system are obtained for different structure parameters. The research results show that this system can implement two types of self-synchronization motion under different structure parameters. The first type fluctuates the phase differences in the proximity of zero rad, while the other one fluctuates the phase differences near π rad. Resonance significantly affects the synchronous motion of the system, rapidly changing in phase differences and amplitudes. An increase in the value of $k_{y2}$ is not conducive to implementing the stable synchronous motion. Finally, the correctness of the theory and electromechanical coupling simulation results is experimentally validated. The results of this paper can be directly used to determine the structural parameters for a new type of vibrating machine.

Appendices

Appendix A

Parameters in Eq. (21)

$$\begin{gathered} a_{x} = \omega_{m}^{4} - \omega_{m}^{2} \left( {\frac{{\eta_{M} }}{{\eta_{k} }}\omega_{x1}^{2} + \beta^{2} \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{x1}^{4} + \eta_{M} \omega_{x1}^{2} + \omega_{x1}^{2} } \right) + \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{x1}^{4} \hfill \\ b_{x} = - \omega_{m}^{3} \left( {\beta \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{x1}^{2} + \beta \eta_{M} \omega_{x1}^{2} + \beta \omega_{x1}^{2} } \right) + 2\beta \omega_{m} \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{x1}^{4} \hfill \\ c_{x1} = - \omega_{m}^{2} + \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{x1}^{2} + \eta_{M} \omega_{x1}^{2} ,\;\;d_{x1} = \beta \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{x1}^{2} \omega_{m} + \beta \eta_{M} \omega_{x1}^{2} \omega_{m} , \hfill \\ d_{x2} = \beta \omega_{x1}^{2} \omega_{m} ,\;\;c_{x2} = \omega_{x1}^{2} ,p_{x} = - \omega_{m}^{2} + \omega_{x1}^{2} \hfill \\ \end{gathered}$$

(32)

$$\begin{gathered} \frac{{\left( {c_{x1} + id_{x1} } \right)}}{{\left( {a_{x} + ib_{x} } \right)}} = \sqrt {\frac{{c_{x1}^{2} + d_{x1}^{2} }}{{a_{x}^{2} + b_{x}^{2} }}} e^{{ - i\theta_{x1} }} ,\;\;\mu_{x11} = \sqrt {\frac{{c_{x1}^{2} + d_{x1}^{2} }}{{a_{x}^{2} + b_{x}^{2} }}} \hfill \\ \frac{{\left( {c_{x2} + id_{x2} } \right)}}{{\left( {a_{x} + ib_{x} } \right)}} = \sqrt {\frac{{c_{x2}^{2} + d_{x2}^{2} }}{{a_{x}^{2} + b_{x}^{2} }}} e^{{ - i\theta_{x2} }} ,\;\;\mu_{x12} = \sqrt {\frac{{c_{x2}^{2} + d_{x2}^{2} }}{{a_{x}^{2} + b_{x}^{2} }}} \hfill \\ \frac{{\left( {p_{x} + id_{x2} } \right)}}{{\left( {a_{x} + ib_{x} } \right)}} = \sqrt {\frac{{p_{x}^{2} + d_{x2}^{2} }}{{a_{x}^{2} + b_{x}^{2} }}} e^{{ - i\theta_{x3} }} ,\;\;\mu_{x22} = \sqrt {\frac{{p_{x}^{2} + d_{x2}^{2} }}{{a_{x}^{2} + b_{x}^{2} }}} \hfill \\ \theta_{x1} = \arctan \frac{{b_{x} c_{x1} - a_{x} d_{x1} }}{{a_{x} c_{x1} + b_{x} d_{x1} }},\;\;\;\;\;\;\;a_{x} c_{x1} + b_{x} d_{x1} > 0 \hfill \\ \theta_{x1} = \pi + \arctan \frac{{b_{x} c_{x1} - a_{x} d_{x1} }}{{a_{x} c_{x1} + b_{x} d_{x1} }},\;\;\;\;a_{x} c_{x1} + b_{x} d_{x1} < 0 \hfill \\ \theta_{x2} = \arctan \frac{{b_{x} c_{x2} - a_{x} d_{x2} }}{{a_{x} c_{x2} + b_{x} d_{x2} }},\;\;\;\;\;\;a_{x} c_{x2} + b_{x} d_{x2} > 0 \hfill \\ \theta_{x2} = \pi + \arctan \frac{{b_{x} c_{x2} - a_{x} d_{x2} }}{{a_{x} c_{x2} + b_{x} d_{x2} }},\;\;\;\;\;a_{x} c_{x2} + b_{x} d_{x2} > 0 \hfill \\ \theta_{x3} = \arctan \frac{{b_{x} p_{x} - a_{x} d_{x2} }}{{a_{x} p_{x} + b_{x} d_{x2} }},\;\;\;\;\;\;a_{x} p_{x} + b_{x} d_{x2} > 0 \hfill \\ \theta_{x3} = \pi + \arctan \frac{{b_{x} p_{x} - a_{x} d_{x2} }}{{a_{x} p_{x} + b_{x} d_{x2} }}\begin{array}{*{20}c} , & {} \\ \end{array} \begin{array}{*{20}c} {} & {} \\ \end{array} a_{x} p_{x} + b_{x} d_{x2} < 0 \hfill \\ \end{gathered}$$

(33)

$$\begin{gathered} A_{x1} = \eta_{m} r\omega_{m}^{2} \sqrt {\mu_{x11}^{2} + \eta_{M}^{2} \mu_{x12}^{2} + 2\eta_{M} \mu_{x11} \mu_{x12} \cos \left[ {\left( {\varphi_{1} - \varphi_{2} } \right) - \theta_{x1} + \theta_{x2} } \right]} \hfill \\ \phi_{x1} = \arctan \frac{{\left( {\mu_{x11} - \eta_{M} \mu_{x12} } \right)\sin \left( {\frac{{\left( {\varphi_{1} - \varphi_{2} } \right) - \theta_{x1} + \theta_{x2} }}{2}} \right)}}{{\left( {\mu_{x11} + \eta_{M} \mu_{x12} } \right)\cos \left( {\frac{{\left( {\varphi_{1} - \varphi_{2} } \right) - \theta_{x1} + \theta_{x2} }}{2}} \right)}}\begin{array}{*{20}c} {} & {} & {} \\ \end{array} \begin{array}{*{20}c} {} & {} \\ \end{array} \left( {\mu_{x11} + \eta_{M} \mu_{x12} } \right)\cos \left( {\frac{{\left( {\varphi_{1} - \varphi_{2} } \right) - \theta_{x1} + \theta_{x2} }}{2}} \right) > 0 \hfill \\ \phi_{x1} = \pi + \arctan \frac{{\left( {\mu_{x11} - \eta_{M} \mu_{x12} } \right)\sin \left( {\frac{{\left( {\varphi_{1} - \varphi_{2} } \right) - \theta_{x1} + \theta_{x2} }}{2}} \right)}}{{\left( {\mu_{x11} + \eta_{M} \mu_{x12} } \right)\cos \left( {\frac{{\left( {\varphi_{1} - \varphi_{2} } \right) - \theta_{x1} + \theta_{x2} }}{2}} \right)}}\begin{array}{*{20}c} {} & {} & {} \\ \end{array} \left( {\mu_{x11} + \eta_{M} \mu_{x12} } \right)\cos \left( {\frac{{\left( {\varphi_{1} - \varphi_{2} } \right) - \theta_{x1} + \theta_{x2} }}{2}} \right) < 0 \hfill \\ \end{gathered}$$

(34)

$$\begin{gathered} A_{x2} = \eta_{m} \eta_{M} r\omega_{m}^{2} \sqrt {\mu_{x12}^{2} + \mu_{x22}^{2} + 2\mu_{x12} \mu_{x22} \cos \left( {\varphi_{1} - \varphi_{2} - \theta_{x2} + \theta_{x3} } \right)} \hfill \\ \phi_{x2} = \arctan \frac{{\left( {\mu_{x12} - \mu_{x22} } \right)\sin \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{x2} + \theta_{x3} }}{2}} \right)}}{{\left( {\mu_{x12} + \mu_{x22} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{x2} + \theta_{x3} }}{2}} \right)}}\begin{array}{*{20}c} {} & {} & {} \\ \end{array} \left( {\mu_{x12} + \mu_{x22} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{x2} + \theta_{x3} }}{2}} \right) > 0 \hfill \\ \phi_{x2} = \pi + \arctan \frac{{\left( {\mu_{x12} - \mu_{x22} } \right)\sin \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{x2} + \theta_{x3} }}{2}} \right)}}{{\left( {\mu_{x12} + \mu_{x22} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{x2} + \theta_{x3} }}{2}} \right)}}\begin{array}{*{20}c} {} & {} & {} \\ \end{array} \left( {\mu_{x12} + \mu_{x22} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{x2} + \theta_{x3} }}{2}} \right) < 0 \hfill \\ \end{gathered}$$

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Appendix B

Parameters in Eq. (22)

$$\begin{gathered} \omega_{y1}^{2} = \frac{{k_{y1} }}{{M_{1} }},\begin{array}{*{20}c} {} & {} \\ \end{array} \frac{{k_{y2} }}{{M_{2} }} = \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{y1}^{2} ,\begin{array}{*{20}c} {} & {} \\ \end{array} \frac{{k_{y1} }}{{M_{2} }} = \frac{{\eta_{M} k_{y1} }}{{M_{1} }} = \eta_{M} \omega_{y1}^{2} \hfill \\ \left[ {\begin{array}{*{20}c} {f_{y1} } \\ {f_{y2} } \\ \end{array} } \right] = \beta \left[ {\begin{array}{*{20}c} {k_{y1} } \\ {k_{y2} } \\ \end{array} } \right],\begin{array}{*{20}c} {} & {} \\ \end{array} \frac{{f_{y1} }}{{M_{1} }} = \beta \omega_{y1}^{2} ,\begin{array}{*{20}c} {} & {} \\ \end{array} \frac{{f_{y2} }}{{M_{2} }} = \beta \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{y1}^{2} {,}\begin{array}{*{20}c} {} & {} \\ \end{array} \frac{{f_{y1} }}{{M_{2} }} = \beta \eta_{M} \omega_{y1}^{2} \hfill \\ \end{gathered}$$

(36)

$$\begin{gathered} a_{y} = \omega_{m}^{4} - \omega_{m}^{2} \left( {\frac{{\eta_{M} }}{{\eta_{k} }}\omega_{y1}^{2} + \beta^{2} \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{y1}^{4} + \eta_{M} \omega_{y1}^{2} + \omega_{y1}^{2} } \right) + \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{y1}^{4} \hfill \\ b_{y} = - \omega_{m}^{3} \left( {\beta \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{y1}^{2} + \beta \eta_{M} \omega_{y1}^{2} + \beta \omega_{y1}^{2} } \right) + 2\beta \omega_{m} \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{y1}^{4} \hfill \\ c_{y1} = - \omega_{m}^{2} + \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{y1}^{2} + \eta_{M} \omega_{y1}^{2} ,d_{y1} = \beta \frac{{\eta_{M} }}{{\eta_{k} }}\omega_{y1}^{2} \omega_{m} + \beta \eta_{M} \omega_{y1}^{2} \omega_{m} , \hfill \\ d_{y2} = \beta \omega_{y1}^{2} \omega_{m} ,c_{y2} = \omega_{y1}^{2} ,p_{y} = - \omega_{m}^{2} + \omega_{y1}^{2} \hfill \\ \end{gathered}$$

(37)

$$\begin{gathered} \theta_{y1} = \arctan \frac{{b_{y} c_{y1} - a_{y} d_{y1} }}{{a_{y} c_{y1} + b_{y} d_{y1} }},\;\mu_{y11} = \sqrt {\frac{{c_{y1}^{2} + d_{y1}^{2} }}{{a_{y}^{2} + b_{y}^{2} }}} \hfill \\ \theta_{y2} = \arctan \frac{{b_{y} c_{y2} - a_{y} d_{y2} }}{{a_{y} c_{y2} + b_{y} d_{y2} }},\;\mu_{y12} = \sqrt {\frac{{c_{y2}^{2} + d_{y2}^{2} }}{{a_{y}^{2} + b_{y}^{2} }}} \hfill \\ \theta_{y3} = \arctan \frac{{b_{y} p_{y} - a_{y} d_{y2} }}{{a_{y} p_{y} + b_{y} d_{y2} }},\;\mu_{y22} = \sqrt {\frac{{p_{y}^{2} + d_{y2}^{2} }}{{a_{y}^{2} + b_{y}^{2} }}} \hfill \\ \end{gathered}$$

(38)

$$\begin{gathered} A_{y1} = \eta_{m} r\omega_{m}^{2} \sqrt {\mu_{y11}^{2} + \eta_{M}^{2} \mu_{y12}^{2} + 2\eta_{M} \mu_{y11} \mu_{y12} \cos \left[ {\varphi_{1} - \varphi_{2} - \theta_{y1} + \theta_{y2} } \right]} \hfill \\ \phi_{y1} = arctan\frac{{\left( {\mu_{y11} - \eta_{M} \mu_{y12} } \right)\sin \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y1} + \theta_{y2} }}{2}} \right)}}{{\left( {\mu_{y11} + \eta_{M} \mu_{y12} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y1} + \theta_{y2} }}{2}} \right)}}\quad \quad \quad \left( {\mu_{y11} + \eta_{M} \mu_{y12} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y1} + \theta_{y2} }}{2}} \right) > 0 \hfill \\ \phi_{y1} = \pi + arctan\frac{{\left( {\mu_{y11} - \eta_{M} \mu_{y12} } \right)\sin \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y1} + \theta_{y2} }}{2}} \right)}}{{\left( {\mu_{y11} + \eta_{M} \mu_{y12} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y1} + \theta_{y2} }}{2}} \right)}}\quad \quad \quad \left( {\mu_{y11} + \eta_{M} \mu_{y12} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y1} + \theta_{y2} }}{2}} \right) < 0 \hfill \\ \end{gathered}$$

(39)

$$\begin{gathered} A_{y2} = \eta_{m} \eta_{M} r\omega_{m}^{2} \sqrt {\mu_{y12}^{2} + \mu_{y22}^{2} + 2\mu_{y12} \mu_{y22} \cos (\varphi_{1} - \varphi_{2} - \theta_{y2} + \theta_{y3} )} \hfill \\ \phi_{y2} = \arctan \frac{{\left( {\mu_{y12} - \mu_{y22} } \right)\sin \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y2} + \theta_{y3} }}{2}} \right)}}{{\left( {\mu_{y12} + \mu_{y22} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y2} + \theta_{y3} }}{2}} \right)}}\quad \quad \quad \left( {\mu_{y12} + \mu_{y22} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y2} + \theta_{y3} }}{2}} \right) > 0 \hfill \\ \phi_{y2} = \pi + \arctan \frac{{\left( {\mu_{y12} - \mu_{y22} } \right)\sin \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y2} + \theta_{y3} }}{2}} \right)}}{{\left( {\mu_{y12} + \mu_{y22} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y2} + \theta_{y3} }}{2}} \right)}}\quad \quad \quad \left( {\mu_{y12} + \mu_{y22} } \right)\cos \left( {\frac{{\varphi_{1} - \varphi_{2} - \theta_{y2} + \theta_{y3} }}{2}} \right) < 0 \hfill \\ \end{gathered}$$

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