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Nonlinear Dynamic Response of a Thin Rectangular Plate Vibration System Excited by a Non-ideal Induction Motor

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Abstract

Purpose

The Sommerfeld effect describes a non-linear jump phenomenon caused by non-ideal sources in a vibration system. The Sommerfeld effect and stability of a thin rectangular plate vibration system excited by a non-ideal induction motor are investigated in this study using an analytical method that combines the average perturbation method and the semi-inverse method.

Methods

The Hamilton principle is applied to construct a non-ideal electromechanical coupling dynamic model. The continuous system model is discretized using the assumed mode method and simplified without taking into account minor cross terms. The exact form of the mode shape function and the transcendental frequency equation of the thin rectangular plate are derived by the semi-inverse method. The system’s steady-state response is achieved by omitting the impact of the motor speed ripple. Meanwhile, the stability of the average speed solution is analyzed with the average perturbation method.

Results

The analytical results are compared to the MATLAB/Simulink simulation results to verify that the proposed method is feasible. Furthermore, the impacts of eccentric masses and motor powers on the dynamic characteristics and stability of the non-ideal vibration system are explored. The results indicate that the speed capture and release may occur around the resonance region. The eccentric masses and motor powers have an impact on the output torque and electromagnetic torque, respectively, which further affects the system’s nonlinear dynamic response.

Conclusion

This paper can serve as a resource for determining the motor power in a vibration system supported by an elastic plate.

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Availability of Data and Materials

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

This study was funded by National Natural Science Foundation of China (Grant No. 51705337), China Postdoctoral Science Foundation (Grant No. 2017M611258), and Natural Science Foundation of Liaoning Province (Grant No. 2019MS245 and LJGD2020011).

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Authors and Affiliations

  1. School of Mechanical Engineering, Shenyang University of Technology, Shenyang, 110870, China

    Wenjie Li, Xiangxi Kong, Qi Xu, Chong Zhou & Ziyu Hao

Authors
  1. Wenjie Li
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  2. Xiangxi Kong
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  3. Qi Xu
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  4. Chong Zhou
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Contributions

All the authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by WL, XK, QX, CZ and ZH. The first draft of the manuscript was written by WL, and all the authors commented on previous versions of the manuscript. All the authors read and approved the final manuscript.

Corresponding author

Correspondence to Xiangxi Kong.

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All the contributing authors have read and consented to the submission of the manuscript in its present form. The authors also declare that they have no conflict of interest.

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Appendix A

Appendix A

The corresponding expressions of Eq. (21)

The differential equation for free vibration of thin plate:

$$D\nabla^{4} w + \rho h\ddot{w} = 0.$$
(A.1)

The form of any order principal vibration is

$$w = \left( {A_{1} \cos (\omega t) + A_{2} \sin (\omega t)} \right)\Phi (x,y).$$
(A.2)

Substituting Eq. (A.2) into Eq. (A.1) and eliminating the time function, we can obtain the eigenvalues equation as follows:

$$\nabla^{4} \Phi (x,y) - \frac{{\omega^{2} \rho h\Phi (x,y)}}{D} = 0$$
(A.3)

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Cite this article

Li, W., Kong, X., Xu, Q. et al. Nonlinear Dynamic Response of a Thin Rectangular Plate Vibration System Excited by a Non-ideal Induction Motor. J. Vib. Eng. Technol. 11, 1211–1227 (2023). https://doi.org/10.1007/s42417-022-00637-2

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  • DOI: https://doi.org/10.1007/s42417-022-00637-2

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