Abstract
In this paper, a newly composite synchronization scheme is proposed to ensure the straight line vibration form of a linear vibration system driven by four exciters. Composite synchronization is a combination of self-synchronization and controlled synchronization. Firstly, controlled synchronization of two pairs of homodromous coupling exciters with zero phase differences is implemented by using the master–slave control structure and the adaptive sliding mode control algorithm. On basis of controlled synchronization, self-synchronization of two coupling exciters rotating in the opposite directions is studied. Based on the perturbation method, the synchronization and stability conditions of composite synchronization are obtained. The theoretical results indicate that composite synchronization of four exciters with zero phase differences can be implemented with different supply frequencies and the straight line vibration form of the linear vibration system also can be obtained. Some simulations are conducted to verify the feasibility of the proposed composite synchronization scheme. The effects of some structural parameters on composite synchronization of four exciters are discussed. Finally, some experiments are operated to validate the effectiveness of the proposed composite synchronization scheme.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China [grant numbers 51705337, 51375080], the China Postdoctoral Science Foundation [grant numbers 2017M611258], the Doctoral Start-up Foundation of Liaoning Province [grant numbers 20170520111] and the Natural Science Foundation of Liaoning Province [grant numbers 2019MS245 and 20180551036].
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Appendices
Appendix 1: [22]
In our work, we take exciters 1 and 2 as example to explain the controlled synchronization of exciters 1, 2 and 3, 4. From Eq. (7), the equations of motion of exciters 1 and 2 are rewritten as
where u1 and u2 are the torque currents of exciters 1 and 2; a1, a2, b1, b2, w1 and w2 are the corresponding coefficients, a1 = -f1/J1, b1 = KT1/J1, w1 = -1/J1, a2 = -f2/J2, b2 = KT2/J2, w2 = -1/J2.
The phase tracking error is expressed as
From Eqs. (27) and (28), we can obtain
where W is a bounded uncertainty, \(W = (a_{1} - a_{2} )\dot{\varphi }_{1} + b_{1} u_{1} + w_{1} T_{L1} - w_{2} T_{L2}\). Here, \(\left| W \right| < \rho\) is assumed, where ρ is a positive constant.
The sliding variable is designed as
where κ and h are two positive constants.
From Eqs. (29) and (30), we can obtain
Choosing \(\dot{S} = 0\) and ignoring the uncertain term W, the equivalent input is obtained as
To reject the parametric perturbations and disturbances, the robust controller is designed
where β is a positive constant and sgn(\(\cdot\)) is the sign function
Thus, the sliding mode controller is obtained as
The adaptive algorithm for the boundary of the uncertainty \(\left| W \right|\) is designed as
where γ is the adaptive gain with positive value. According to Eq. (35), the switch gain β is deduced as
The estimated switch gain is defined as \(\hat{\beta }\), where \(\hat{\beta } > \rho\). Moreover, the estimated error \(\hat{\beta }\) can be expressed as
By using the modified method [24], the modified adaptive switch gain is obtained as
with
where p and δ are two small positive constants.
In order more to reduce the chattering phenomenon, the saturated function \({\text{sat}}( \cdot )\) is employed to replace the sign function \(\text{sgn} ( \cdot )\):
where ϑ is a positive constant. Finally, the controller u2 is expressed as
Appendix 2: The terms of Eq. (22)
withª
where Te01 = Te1(ω0), Te04 = Te4(ω0), ke01 and ke04 are calculated by substituting parameter values of motors in the flowing equation
Appendix 3: The terms of Eq. (26)
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Kong, X., Zhou, C. & Wen, B. Composite synchronization of four exciters driven by induction motors in a vibration system. Meccanica 55, 2107–2133 (2020). https://doi.org/10.1007/s11012-020-01246-7
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DOI: https://doi.org/10.1007/s11012-020-01246-7