Sommerfeld effect and synchronization analysis in a simply supported beam system excited by two non-ideal induction motors

Abstract

In this paper, Sommerfeld effect and self-synchronization of two non-ideal induction motors in a simply supported beam system are studied. Based on fully considering non-ideal characteristics of induction motors and interactions between the beam structure and induction motors, a continuous model is developed and discretized by using the assumed mode method. A typical electromechanical coupling dynamic model of the vibration system is formulated to perform an investigation on its dynamic behaviors and stability. Synchronization and stability analysis are implemented by using the perturbation method. The numerical simulation is used to confirm the feasibility of the analytical approach. The effects of angular speeds, structure parameters and power supply frequencies on Sommerfeld effect and synchronous motion of two induction motors are analyzed.

Appendix A

The terms of Eq. (28)

$$ p_{11} = \eta_{1}^{2} m_{0} \sum\limits_{j = 1}^{n} {\frac{{\psi_{j}^{2} (x_{1} )\cos \gamma_{j} }}{{2\mu_{j} }}} , $$

(A.1)

$$ p_{12} = \eta_{1} \eta_{2} m_{0} \sum\limits_{j = 1}^{n} {\frac{{\psi_{j} (x_{1} )\psi_{j} (x_{2} )\cos \left( { - \theta_{0} - \gamma_{j} } \right)}}{{2\mu_{j} }}} , $$

(A.2)

$$ p_{21} = \eta_{1} \eta_{2} m_{0} \sum\limits_{j = 1}^{n} {\frac{{\psi_{j} (x_{1} )\psi_{j} (x_{2} )\cos \left( {\theta_{0} - \gamma_{j} } \right)}}{{2\mu_{j} }}} , $$

(A.3)

$$ p_{22} = m_{0} \eta_{2}^{2} \sum\limits_{j = 1}^{n} {\frac{{\psi_{j}^{2} (x_{2} )\cos \gamma_{j} }}{{2\mu_{j} }}} , $$

(A.4)

$$ q_{11} = - m_{0} \eta_{1}^{2} \varOmega \sum\limits_{j = 1}^{n} {\frac{{\psi_{j}^{2} (x_{1} )\sin ( - \gamma_{j} )}}{{\mu_{j} }}} , $$

(A.5)

$$ q_{12} = - m_{0} \eta_{1} \eta_{2} \varOmega \sum\limits_{j = 1}^{n} {\frac{{\psi_{j} (x_{1} )\psi_{j} (x_{2} )\sin \left( { - \theta_{0} - \gamma_{j} } \right)}}{{\mu_{j} }}} , $$

(A.6)

$$ q_{13} = m_{0} \eta_{1} \eta_{2} \varOmega \sum\limits_{j = 1}^{n} {\frac{{\psi_{j} (x_{1} )\psi_{j} (x_{2} )\cos \left( { - \theta_{0} - \gamma_{j} } \right)}}{{2\mu_{j} }}} , $$

(A.7)

$$ q_{21} = - m_{0} \eta_{1} \eta_{2} \varOmega \sum\limits_{j = 1}^{n} {\frac{{\psi_{j} (x_{1} )\psi_{j} (x_{2} )\sin \left( {\theta_{0} - \gamma_{j} } \right)}}{{\mu_{j} }}} , $$

(A.8)

$$ q_{22} = - m_{0} \eta_{2}^{2} \varOmega \sum\limits_{j = 1}^{n} {\frac{{\psi_{j}^{2} (x_{2} )\sin ( - \gamma_{j} )}}{{\mu_{j} }}} , $$

(A.9)

$$ q_{23} = - m_{0} \eta_{1} \eta_{2} \varOmega \sum\limits_{j = 1}^{n} {\frac{{\psi_{i} (x_{1} )\psi_{i} (x_{2} )\cos \left( {\theta_{0} - \gamma_{j} } \right)}}{{2\mu_{j} }}} , $$

(A.10)

$$ \chi_{1} = - m_{0} \eta_{1} \varOmega \sum\limits_{j = 1}^{n} {\frac{{\psi_{j} (x_{1} )\left( {\eta_{1} \psi_{j} (x_{1} )\sin ( - \gamma_{j} ) + \eta_{2} \psi_{j} (x_{2} )\sin ( - \theta_{0} - \gamma_{j} )} \right)}}{{2\mu_{j} }}} , $$

(A.11)

$$ \chi_{2} = - m_{0} \eta_{2} \varOmega \sum\limits_{j = 1}^{n} {\frac{{\psi_{j} (x_{2} )\left( {\eta_{2} \psi_{j} (x_{2} )\sin ( - \gamma_{j} ) + \eta_{1} \psi_{j} (x_{1} )\sin (\theta_{0} - \gamma_{j} )} \right)}}{{2\mu_{j} }}} , $$

(A.12)

$$ \begin{aligned} k_{e} & = - \varOmega \frac{{\partial T_{e} }}{\partial \omega }(\varOmega ) \\ & = \frac{{6\varOmega n_{p}^{2} V^{2} R_{r}^{2} \left( {R_{s} + 2\pi R_{r} f_{s} /(2\pi f_{s} - n_{p} \varOmega )} \right)2\pi f_{s} }}{{(2\pi f_{s} - n_{p} \varOmega )^{3} \left( {\left( {R_{s} + 2\pi R_{r} f_{s} /(2\pi f_{s} - n_{p} \varOmega )} \right)^{2} + 2\pi f_{s}^{2} \left( {L_{1s} + L_{1r} } \right)^{2} } \right)^{2} }} \\ & \quad - \frac{{3\varOmega n_{p} V^{2} R_{r} }}{{(2\pi f_{s} - n_{p} \varOmega )^{2} \left( {\left( {R_{s} + R_{r} 2\pi f_{s} /(2\pi f_{s} - n_{p} \varOmega )} \right)^{2} + 2\pi f_{s}^{2} \left( {L_{1s} + L_{1r} } \right)^{2} } \right)}}, \\ \end{aligned} $$

(A.13)

where k_e10 and k_e20 can be obtained by substituting the parameters of motors 1 and 2 into Eq. (A.13), respectively.