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Study on electromechanical coupling torsional resonance characteristics of gear system driven by PMSM: a case on shearer semi-direct drive cutting transmission system

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Abstract

In this paper, the electromechanical coupling torsional resonance characteristics of the multistage gear transmission system driven by the low-speed and high-power permanent magnet synchronous motor (PMSM) are studied. Considering the PMSM electromagnetic effect and bending-torsional vibration characteristics, the mechanical-electromagnetic coupling dynamic model suitable for the speed change process is established. Then, the PMSM electromagnetic stiffness and damping characteristics are analyzed through theoretical derivation. On the basis of electromagnetic stiffness, the influence law of electromagnetic effect on the gear transmission system natural frequency and vibration characteristics is revealed systematically. By both of Campbell diagram and modal energy distribution, the potential resonance points under the influence of internal and external excitation are initially acquired. Ultimately, the potential resonance points are identified by frequency sweep analysis and time domain simulation, and the energy concentration components at resonance point are further analyzed. The results show that the electromagnetic effect reduces the first-order natural frequency and has little influence on others. Moreover, the semi-direct drive cutting transmission system in this paper has the resonance risk caused by the gear time-varying meshing stiffness when the PMSM speed is close to 330 r/min, which should be paid more attention in actual operation. This study can provide some guidance for the design and speed regulation of the gear drive system driven by high-power PMSM.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 51775543), the Key Research and Development Project of Xuzhou (Grant No. KC17014) and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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Authors and Affiliations

  1. School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou, 221116, China

    Song Jiang, Wei Li, Yuqiao Wang, Xuefeng Yang & Shaoyi Xu

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  1. Song Jiang
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  2. Wei Li
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  5. Shaoyi Xu
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Correspondence to Wei Li.

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Appendix

Appendix

The mass matrix and total stiffness coefficient matrix of gear system:

$$ {\text{M}} = diag\left[ {\begin{array}{*{20}c} {m_{1} } & {m_{2} } & {m_{3} } & {m_{4} } & {m_{5} } & {m_{6} } & {m_{7} } & {m_{8} } & {J_{m} } & {J_{1} } & {J_{2} } & {J_{3} } & {J_{4} } & {J_{5} } & {J_{6} } & {J_{7} } & {J_{8} } & {J_{L} } \\ \end{array} } \right] $$
$$ {\text{K}}_{{\text{q}}} = \left[ {\begin{array}{*{20}c} {{\text{K}}_{{{\text{q}}1}} } & {{\text{K}}_{{{\text{q}}2}} } \\ {{\text{K}}_{{{\text{q}}3}} } & {{\text{K}}_{{{\text{q}}4}} } \\ \end{array} } \right] $$
$$ {\text{K}}_{{{\text{q}}1}} = \left[ {\begin{array}{*{20}c} {K_{1} + K_{12} } & { - K_{12} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ { - K_{12} } & {K_{2} + K_{12} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {K_{3} + K_{34} } & { - K_{34} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - K_{34} } & {K_{4} + K_{34} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {K_{5} + K_{56} } & { - K_{56} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - K_{56} } & {K_{6} + K_{56} + K_{67} } & { - K_{67} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - K_{67} } & {K_{7} + K_{67} + K_{78} } & { - K_{78} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - K_{78} } & {K_{8} + K_{78} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {K_{a} } \\ \end{array} } \right] $$
$$ {\text{K}}_{{{\text{q}}2}} = \left[ {\begin{array}{*{20}c} {K_{12} R_{1} } & { - K_{12} R_{2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ { - K_{12} R_{1} } & {K_{12} R_{2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {K_{34} R_{3} } & { - K_{34} R_{4} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - K_{34} R_{3} } & {K_{34} R_{4} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {K_{56} R_{5} } & { - K_{56} R_{6} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - K_{56} R_{5} } & {K_{56} R_{6} + K_{67} R_{6} } & { - K_{67} R_{7} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - K_{67} R_{6} } & {K_{67} R_{7} + K_{78} R_{7} } & { - K_{78} R_{8} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - K_{78} R_{7} } & {K_{78} R_{8} } & 0 \\ { - K_{a} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] $$
$$ {\text{K}}_{{{\text{q}}3}} = \left[ {\begin{array}{*{20}c} {K_{12} R_{1} } & { - K_{12} R_{1} } & 0 & 0 & 0 & 0 & 0 & 0 & { - K_{a} } \\ { - K_{12} R_{2} } & {K_{12} R_{2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {K_{34} R_{3} } & { - K_{34} R_{3} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - K_{34} R_{4} } & {K_{34} R_{4} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {K_{56} R_{5} } & { - K_{56} R_{5} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - K_{56} R_{6} } & {K_{56} R_{6} + K_{67} R_{6} } & { - K_{67} R_{6} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - K_{67} R_{7} } & {K_{67} R_{7} + K_{78} R_{7} } & { - K_{78} R_{7} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - K_{78} R_{8} } & {K_{78} R_{8} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] $$
$$ K_{q4} = \left[ {\begin{array}{*{20}c} {K_{a} + K_{12} R_{1}^{2} } & { - K_{12} R_{1} R_{2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ { - K_{12} R_{1} R_{2} } & {K_{b} + K_{12} R_{2}^{2} } & { - K_{b} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - K_{b} } & {K_{b} + K_{34} R_{3}^{2} } & { - K_{34} R_{3} R_{4} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - K_{34} R_{3} R_{4} } & {K_{c} + K_{34} R_{4}^{2} } & { - K_{c} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - K_{c} } & {K_{c} + K_{56} R_{5}^{2} } & { - K_{56} R_{5} R_{6} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - K_{56} R_{5} R_{6} } & {K_{56} R_{6}^{2} + K_{67} R_{6}^{2} } & { - K_{67} R_{6} R_{7} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {K_{67} R_{7}^{2} + K_{78} R_{8}^{2} } & { - K_{78} R_{7} R_{8} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - K_{78} R_{7} R_{8} } & {K_{d} + K_{78} R_{8}^{2} } & { - K_{d} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - K_{d} } & {K_{d} } \\ \end{array} } \right] $$

In some important simulation parameters, the gear meshing stiffness is the average meshing stiffness in the table:

Symbol

Value

Unit

Symbol

Value

Unit

Kpω

20

 

C34

2.82e4

N·s/m·rad

Kiω

50

 

C56

4.26e4

N·s/m·rad

Kpq

15

 

C67

5.44e4

N·s/m·rad

Kiq

20.75

 

C78

6.39e4

N·s/m·rad

Ka

1.10e6

N/m·rad

m1

6.51

kg

Kb

2.34e7

N/m·rad

m2

56.2

kg

Kc

2.85e7

N/m·rad

m3

13.02

kg

Kd

1.79e8

N/m·rad

m4

87.26

kg

Ca

4.43e2

N·s/m·rad

m5

27.74

kg

Cb

1.73e3

N·s/m·rad

m6/m7

60.50

kg

Cc

2.72e3

N·s/m·rad

m8

131.38

kg

Cd

5.64e3

N/m·rad

J1

0.024

kg m2

K12

2.78e9

N/m·rad

J2

0.82

kg m2

K34

2.87e9

N/m·rad

J3

0.089

kg m2

K56

3.68e9

N/m·rad

J4

2.034

kg m2

K67

4.02e9

N/m·rad

J5

0.26

kg m2

K78

4.18e9

N/m·rad

J6/J7

1.18

kg m2

C12

2.02e4

N·s/m·rad

J8

3.26

kg m2

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Jiang, S., Li, W., Wang, Y. et al. Study on electromechanical coupling torsional resonance characteristics of gear system driven by PMSM: a case on shearer semi-direct drive cutting transmission system. Nonlinear Dyn 104, 1205–1225 (2021). https://doi.org/10.1007/s11071-021-06364-9

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