Self-Synchronization and Vibration Isolation Theories in Anti-resonance System with Dual-Motor and Double-Frequency Actuation

Abstract

Background

Traditional vibrating system has the characteristics of poor isolation ability and easy to plug the screen.

Purpose

To enhance vibration isolation ability and probability of screen plugging during the operation of vibrating screens, an anti-resonance system with dual-motor and double-frequency actuation is proposed.

Method

For mastering the dynamic characteristics of the system, the self-synchronization mechanism and stability of the eccentric rotors bonded in the induction motor is explored. First, the dynamic equation of the system is established using Lagrange method. Subsequently, the synchronization criterion between the eccentric rotors is studied by small parameter method. In light of Hamilton principle, the stability of the synchronous states of the rotors is determined. Besides, the influence of the structural parameters on vibration isolation ability and synchronization characteristics is discussed by numerical computation. Finally, according to Runge-Kutta principle, some simulation results are obtained to verify the validity of the self-synchronization and vibration isolation in this paper.

Results and conclusions

The results show that the synchronous state of the eccentric rotors is mainly effected by installation angle of the motors; the ability of the vibration isolation is mostly determined by spring stiffness and rotor mass. The present findings should give theoretical direction to design the anti-resonance system with double-frequency actuation.

Appendices

Appendix A

The parameters of Eq. (13)

$$ \begin{aligned} & \eta _{1} = m_{1} {\text{/}}M,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \eta _{2} = m_{{\text{2}}} {\text{/}}M{\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \eta _{3} = M_{1} /M{\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \eta _{4} = m_{1} /M_{1} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \eta _{5} = m_{2} /M_{1} \\ & \eta _{{12}} = m_{1} /m_{2} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} r_{0} = r/l_{e} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} r_{l} = l/l_{e} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l_{e} = \sqrt {J_{2} /M} \\ & \omega _{{x_{1} }} = \sqrt {k_{{x_{1} }} /M_{1} } {\kern 1pt} {\kern 1pt} {\kern 1pt} ,\xi _{{x_{1} }} = f_{{x_{1} }} /2\sqrt {M_{1} k_{{x_{1} }} } {\kern 1pt} {\kern 1pt} {\kern 1pt} ,n_{{VLx_{1} }} = \omega _{m} /\omega _{{x_{1} }} ,n_{{VHx_{1} }} = 2\omega _{m} /\omega _{{x_{1} }} \\ & \omega _{{x_{2} }} = \sqrt {k_{{x_{2} }} /M} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \xi _{{x_{2} }} = f_{{x_{2} }} /2\sqrt {Mk_{{x_{2} }} } {\kern 1pt} {\kern 1pt} {\kern 1pt} ,n_{{ILx_{2} }} = \omega _{m} /\omega _{{x_{2} }} ,n_{{IHx_{2} }} = 2\omega _{m} /\omega _{{x_{2} }} \\ & \omega _{{y_{1} }} = \sqrt {k_{{y_{1} }} /M_{1} } {\kern 1pt} {\kern 1pt} {\kern 1pt} ,\xi _{{y_{1} }} = f_{{y_{1} }} /2\sqrt {M_{1} k_{{y_{1} }} } {\kern 1pt} {\kern 1pt} {\kern 1pt} ,n_{{VLy_{1} }} = \omega _{m} /\omega _{{y_{1} }} ,n_{{VHy_{1} }} = 2\omega _{m} /\omega _{{y_{1} }} \\ & \omega _{{y_{2} }} = \sqrt {k_{{y_{2} }} /M} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \xi _{{y_{2} }} = f_{{y_{2} }} /2\sqrt {Mk_{{y_{2} }} } {\kern 1pt} {\kern 1pt} {\kern 1pt} ,n_{{ILy_{2} }} = \omega _{m} /\omega _{{y_{2} }} ,n_{{IHy_{2} }} = 2\omega _{m} /\omega _{{y_{2} }} \\ & \omega _{{\psi _{1} }} = \sqrt {k_{{\psi _{1} }} /M_{1} } {\kern 1pt} {\kern 1pt} {\kern 1pt} ,\xi _{{\psi _{1} }} = f_{{\psi _{1} }} /2\sqrt {M_{1} k_{{\psi _{1} }} } {\kern 1pt} {\kern 1pt} {\kern 1pt} ,n_{{VL\psi _{1} }} = \omega _{m} /\omega _{{\psi _{1} }} ,n_{{VH\psi _{1} }} = 2\omega _{m} /\omega _{{\psi _{1} }} \\ & \omega _{{\psi _{2} }} = \sqrt {k_{{\psi _{2} }} /M} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \xi _{{\psi _{2} }} = f_{{\psi _{2} }} /2\sqrt {Mk_{{\psi _{2} }} } {\kern 1pt} {\kern 1pt} {\kern 1pt} ,n_{{IL\psi _{2} }} = \omega _{m} /\omega _{{\psi _{2} }} ,n_{{IH\psi _{2} }} = 2\omega _{m} /\omega _{{\psi _{1} }} , \\ \end{aligned} $$

(31)

$$ \begin{gathered} \mu _{{Ij^{ * } j_{2} }} = \sqrt {\frac{{a_{{Ij^{ * } j_{2} }}^{2} + b_{{j^{ * } }}^{2} }}{{c_{{j^{ * } }}^{2} + d_{{j^{ * } }}^{2} }}} \quad \gamma _{{Ij^{ * } j_{2} }} = \arctan \frac{{a_{{Ij^{ * } j_{2} }} d_{{j^{ * } }} - b_{{j^{ * } }} c_{{j^{ * } }} }}{{a_{{Ij^{ * } j_{2} }} c_{{j^{ * } }} + b_{{j^{ * } }} d_{{j^{ * } }} }}, \hfill \\ \mu _{{Vj^{ * } j_{1} }} = \sqrt {\frac{{a_{{Vj^{ * } j_{1} }}^{2} + b_{{j^{ * } }}^{2} }}{{c_{{j^{ * } }}^{2} + d_{{j^{ * } }}^{2} }}} \quad \gamma _{{Vj^{ * } j_{1} }} = \arctan \frac{{a_{{Vj^{ * } j_{1} }} d_{{j^{ * } }} - b_{{j^{ * } }} c_{{j^{ * } }} }}{{a_{{Vj^{ * } j_{1} }} c_{{j^{ * } }} + b_{{j^{ * } }} d_{{j^{ * } }} }}, \hfill \\ \end{gathered} $$

(32)

$$ \begin{aligned} & a_{{Ij^{ * } j_{2} }} = n_{{Ij^{ * } j_{2} }}^{2} (1 - n_{{Vj^{ * } j_{1} }}^{2} ), \\ & a_{{Vj^{ * } j_{1} }} = n_{{Ij^{ * } j_{2} }}^{2} , \\ & b_{{j^{ * } }} = 2\xi _{{j_{1} }} n_{{Vj^{ * } j_{1} }} n_{{Ij^{ * } j_{2} }}^{2} , \\ & c_{{j^{ * } }} = (1 - n_{{Vj^{ * } j_{1} }}^{2} )(1 - n_{{Ij^{ * } j_{2} }}^{2} ) - n_{{Ij^{ * } j_{2} }} (4\xi _{{j_{2} }} \xi _{{j_{1} }} n_{{Vj^{ * } j_{1} }} + \eta _{3} n_{{Ij^{ * } j_{2} }} ), \\ & d_{{j^{ * } }} = 2[\xi _{{j_{1} }} n_{{Vj^{ * } j_{1} }} + \xi _{{j_{2} }} n_{{Ij^{ * } j_{2} }} (1 - n_{{Vj^{ * } j_{1} }}^{2} ) - (\eta _{3} + 1)\xi _{{j_{1} }} n_{{Ij^{ * } j_{2} }}^{2} n_{{Vj^{ * } j_{1} }} ], \\ \end{aligned} $$

(33)

where, j^* = H, L; j = x, y, ψ

The parameters of Eqs. (17) and (18)

$$ a_{{11}}^{*} = m_{1} r^{2} \omega _{{m0}} W_{{c1}} , $$

(34)

$$ a_{{12}}^{*} = m_{1} r^{2} \omega _{{m0}} W_{{c1}} , $$

(35)

$$ a_{{21}}^{*} = \frac{1}{2}m_{2} r^{2} \omega _{{m0}} W_{{c2}} , $$

(36)

$$ a_{{22}}^{*} = - \frac{1}{2}m_{2} r^{2} \omega _{{m0}} W_{{c2}} , $$

(37)

$$ a_{{11}} = m_{1} r^{2} \omega _{{\omega 0}}^{2} \left[ {4W_{{s1}} + \frac{1}{2}H_{1} \sin (\bar{\alpha }_{0} + 2\beta _{2} - \beta _{1} + 2\gamma _{{IL\psi _{2} }} )} \right], $$

(38)

$$ a_{{12}} = m_{1} r^{2} \omega _{{\omega 0}}^{2} \left[ {4W_{{s1}} - \frac{1}{2}H_{1} \sin (\bar{\alpha }_{0} + 2\beta _{2} - \beta _{1} + 2\gamma _{{IL\psi _{2} }} )} \right]{\kern 1pt} , $$

(39)

$$ {\kern 1pt} a_{{21}} = m_{2} r^{2} \omega _{{\omega 0}}^{2} [W_{{s2}} + 2H\sin (\bar{\alpha }_{0} + {\text{2}}\beta _{2} - \beta _{1} + \gamma _{{IL\psi _{2} }} - \gamma _{{IH\psi _{2} }} )], $$

(40)

$$ a_{{22}} = - m_{2} r^{2} \omega _{{\omega 0}}^{2} W_{{s2}} , $$

(41)

$$ a_{1} = \frac{1}{2}{\kern 1pt} m_{1} r^{2} \omega _{{\omega 0}}^{2} \left[ {4W_{{s1}} + {\kern 1pt} \frac{1}{2}H_{1} \sin (\bar{\alpha }_{0} + 2\beta _{2} - \beta _{1} + 2\gamma _{{IL\psi _{2} }} )} \right], $$

(42)

$$ a_{2} = \frac{1}{2}m_{2} r^{2} \omega _{{\omega 0}}^{2} [W_{{s2}} + 2H_{2} \sin (\bar{\alpha }_{0} + {\text{2}}\beta _{2} - \beta _{1} + \gamma _{{IL\psi _{2} }} - \gamma _{{IH\psi _{2} }} )], $$

(43)

$$ W_{{c1}} = \eta _{1} \mu _{{IHx_{2} }} \cos \gamma _{{IHx_{2} }} + \eta _{1} \mu _{{IHy_{2} }} cos\gamma _{{IHy_{2} }} + \eta _{1} r_{l}^{2} \mu _{{IH\psi _{2} }} \cos \gamma _{{IH\psi _{2} }} {\kern 1pt} , $$

(44)

$$ W_{{s1}} = \eta _{1} \mu _{{IHx_{2} }} \sin \gamma _{{IHx_{2} }} + \eta _{1} \mu _{{IHy_{2} }} \sin \gamma _{{IHy_{2} }} + \eta _{1} \mu _{{IH\psi _{2} }} r_{l}^{2} sin\gamma _{{IH\psi _{2} }} , $$

(44)

$$ W_{{c2}} = \eta _{2} \mu _{{ILx_{2} }} \cos \gamma _{{ILx_{2} }} + \eta _{2} \mu _{{ILy_{2} }} \cos \gamma _{{ILy_{2} }} + r_{l}^{2} \eta _{2} \mu _{{IL\psi _{2} }} \cos \gamma _{{IL\psi _{2} }} , $$

(45)

$$ W_{{s2}} = \eta _{2} \mu _{{ILx_{2} }} \sin \gamma _{{ILx_{2} }} + \eta _{2} \mu _{{ILy_{2} }} \sin \gamma _{{ILy_{2} }} + r_{l}^{2} \eta _{2} \mu _{{IL\psi _{2} }} \sin \gamma _{{IL\psi _{2} }} , $$

(46)

$$ H_{1} = r_{0} r_{l}^{3} \eta _{2}^{2} \mu _{{IL\psi _{2} }}^{2} , $$

(47)

$$ H_{2} = r_{0} r_{l}^{3} \eta _{1} \eta _{2} \mu _{{IH\psi _{2} }} \mu _{{IL\psi _{2} }} . $$

(48)

Appendix B

The parameters of Eq. (20)

$$ {\mathbf{B}} = \left[ \begin{gathered} b_{{11}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b_{{12}} \hfill \\ b_{{21}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b_{{22}} \hfill \\ \end{gathered} \right],{\mathbf{C}} = \left[ \begin{gathered} c_{{11}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} c_{{12}} \hfill \\ c_{{21}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} c_{{22}} \hfill \\ \end{gathered} \right],{\mathbf{E}} = \left[ \begin{gathered} e_{1} \hfill \\ e_{2} \hfill \\ \end{gathered} \right],{\kern 1pt} {\mathbf{\dot{\bar{\varepsilon }}}} = \left[ \begin{gathered} \dot{\bar{\varepsilon }}_{1} \hfill \\ \dot{\bar{\varepsilon }}_{2} \hfill \\ \end{gathered} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathbf{\bar{\varepsilon }}} = \left[ \begin{gathered} \bar{\varepsilon }_{1} \hfill \\ \bar{\varepsilon }_{2} \hfill \\ \end{gathered} \right]{\kern 1pt} , $$

(49)

$$ b_{{11}} = 2J_{{01}} \omega _{{m0}} + J_{{02}} \omega _{{m0}} + a_{{11}}^{*} + a_{{21}}^{*} {\kern 1pt} , $$

(50)

$$ b_{{12}} = 2J_{{01}} \omega _{{m0}} - J_{{02}} \omega _{{m0}} + a_{{12}}^{*} + a_{{22}}^{*} , $$

(51)

$$ b_{{21}} = 2J_{{01}} \omega _{{m0}} - J_{{02}} \omega _{{m0}} + a_{{11}}^{*} - a_{{21}}^{*} , $$

(52)

$$ b_{{22}} = 2J_{{01}} \omega _{{m0}} + J_{{02}} \omega _{{m0}} + a_{{12}}^{*} - a_{{22}}^{*} , $$

(53)

$$ c_{{11}} = - k_{{e01}} - k_{{e02}} - a_{{11}} - a_{{21}} - 2f_{1} \omega _{{m0}} - f_{2} \omega _{{m0}} , $$

(54)

$$ c_{{12}} = {\kern 1pt} - k_{{e01}} + k_{{e02}} - a_{{12}} - a_{{22}} - 2f_{1} \omega _{{m0}} + f_{2} \omega _{{m0}} , $$

(55)

$$ c_{{21}} = {\kern 1pt} - k_{{e01}} + k_{{e02}} - a_{{11}} + a_{{21}} - 2f_{1} \omega _{{m0}} + f_{2} \omega _{{m0}} , $$

(56)

$$ c_{{22}} = - k_{{e01}} - k_{{e02}} - a_{{12}} + a_{{22}} - 2f_{1} \omega _{{m0}} - f_{2} \omega _{{m0}} , $$

(57)

$$ e_{1} = T_{{e01}} + T_{{e02}} - 2f_{1} \omega _{{m0}} - f_{2} \omega _{{m0}} - a_{1} - a_{2} , $$

(58)

$$ e_{2} = T_{{e01}} - T_{{e02}} - 2f_{1} \omega _{{m0}} + f_{2} \omega _{{m0}} - a_{1} + a_{2} . $$

(59)

The parameters of Eq. (21)

$$ N_{{c0}} = \sqrt {a_{{c0}}^{2} + b_{{c0}}^{2} } , $$

(60)

$$ N_{{s0}} = \sqrt {a_{{s0}}^{2} + b_{{s0}}^{2} } , $$

(61)

$$ N_{{c1}} = \sqrt {a_{{c1}}^{2} + b_{{c1}}^{2} } , $$

(62)

$$ N_{{s1}} = \sqrt {a_{{s1}}^{2} + b_{{s1}}^{2} } , $$

(63)

$$ \theta _{{s0}} = \left\{ \begin{gathered} \arctan (b_{{s0}} /a_{{s0}} ),a_{{s0}} {\kern 1pt} > 0 \hfill \\ \pi + \arctan (b_{{s0}} /a_{{s0}} ),a_{{s0}} {\kern 1pt} < 0{\kern 1pt} \hfill \\ \end{gathered} \right., $$

(64)

$$ \theta _{{c0}} = \left\{ \begin{gathered} \arctan (b_{{c0}} /a_{{c0}} ),a_{{c0}} {\kern 1pt} > 0 \hfill \\ \pi + \arctan (b_{{c0}} /a_{{c0}} ),a_{{c0}} {\kern 1pt} < 0{\kern 1pt} \hfill \\ \end{gathered} \right.{\kern 1pt} {\kern 1pt} , $$

(65)

$$ \theta _{{s0}} = \left\{ \begin{gathered} \arctan (b_{{s1}} /a_{{s1}} {\kern 1pt} ),a_{{s1}} {\kern 1pt} > 0 \hfill \\ \pi + \arctan (b_{{s1}} /a_{{s1}} {\kern 1pt} ),a_{{s1}} < 0{\kern 1pt} \hfill \\ \end{gathered} \right., $$

(66)

$$ \theta _{{c0}} = \left\{ \begin{gathered} \arctan (b_{{c1}} /a_{{c1}} {\kern 1pt} ),a_{{c1}} {\kern 1pt} > 0 \hfill \\ \pi + \arctan {\kern 1pt} (b_{{c1}} /a_{{c1}} {\kern 1pt} ),a_{{c1}} {\kern 1pt} < 0{\kern 1pt} \hfill \\ \end{gathered} \right.{\kern 1pt} {\kern 1pt} , $$

(67)

$$ a_{{c0}} = \left[ {\frac{1}{4}\eta _{{12}} H_{1} \cos \left( {2\beta _{2} - \beta _{1} } \right)\cos \left( {2\gamma _{{IL\psi _{2} }} } \right) - H_{2} \cos \left( {{\text{2}}\beta _{2} - \beta _{1} } \right)\cos \left( {\gamma _{{IL\psi _{2} }} - \gamma _{{IH\psi _{2} }} } \right)} \right], $$

(68)

$$ b_{{c0}} = \left[ {\frac{1}{4}\eta _{{12}} H_{1} \sin \left( {2\beta _{2} - \beta _{1} } \right)\cos \left( {2\gamma _{{IL\psi _{2} }} } \right) - H_{2} \sin \left( {{\text{2}}\beta _{2} - \beta _{1} } \right)\cos \left( {\gamma _{{IL\psi _{2} }} - \gamma _{{IH\psi _{2} }} } \right)} \right], $$

(69)

$$ a_{{s0}} = - \left[ {\frac{1}{4}\eta _{{12}} H_{1} \sin \left( {2\beta _{2} - \beta _{1} } \right)\sin \left( {2\gamma _{{IL\psi _{2} }} } \right) - H_{2} \sin \left( {{\text{2}}\beta _{2} - \beta _{1} } \right)\sin \left( {\gamma _{{IL\psi _{2} }} - \gamma _{{IH\psi _{2} }} } \right)} \right], $$

(70)

$$ b_{{s0}} = \left[ {\frac{1}{4}\eta _{{12}} H_{1} \cos \left( {2\beta _{2} - \beta _{1} } \right)\sin \left( {2\gamma _{{IL\psi _{2} }} } \right) - H_{2} \cos \left( {{\text{2}}\beta _{2} - \beta _{1} } \right)\sin \left( {\gamma _{{IL\psi _{2} }} - \gamma _{{IH\psi _{2} }} } \right)} \right], $$

(71)

$$ a_{{c1}} = \left[ {\frac{1}{4}\eta _{{12}} H_{1} \cos \left( {2\beta _{2} - \beta _{1} } \right)\cos \left( {2\gamma _{{IL\psi _{2} }} } \right) + H_{2} \cos \left( {{\text{2}}\beta _{2} - \beta _{1} } \right)\cos \left( {\gamma _{{IL\psi _{2} }} - \gamma _{{IH\psi _{2} }} } \right)} \right], $$

(72)

$$ b_{{c1}} = \left[ {\frac{1}{4}\eta _{{12}} H_{1} \sin \left( {2\beta _{2} - \beta _{1} } \right)\cos \left( {2\gamma _{{IL\psi _{2} }} } \right) + H_{2} \sin \left( {{\text{2}}\beta _{2} - \beta _{1} } \right)\cos \left( {\gamma _{{IL\psi _{2} }} - \gamma _{{IH\psi _{2} }} } \right)} \right], $$

(73)

$$ a_{{s1}} = - \left[ {\frac{1}{4}\eta _{{12}} H_{1} \sin \left( {2\beta _{2} - \beta _{1} } \right)\sin \left( {2\gamma _{{IL\psi _{2} }} } \right) + H_{2} \sin \left( {{\text{2}}\beta _{2} - \beta _{1} } \right)\sin \left( {\gamma _{{IL\psi _{2} }} - \gamma _{{IH\psi _{2} }} } \right)} \right], $$

(74)

$$ b_{{s1}} = \left[ {\frac{1}{4}\eta _{{12}} H_{1} \cos \left( {2\beta _{2} - \beta _{1} } \right)\sin \left( {2\gamma _{{IL\psi _{2} }} } \right) + H_{2} \cos \left( {{\text{2}}\beta _{2} - \beta _{1} } \right)\sin \left( {\gamma _{{IL\psi _{2} }} - \gamma _{{IH\psi _{2} }} } \right)} \right]. $$

(75)