Continuous Rotor Dynamics of Multi-Disc and Multi-Span Rotor: A Theoretical and Numerical Investigation on the Continuous Model and Analytical Solution for Unbalance Responses
Abstract
:1. Introduction
2. Theory
2.1. Continous Model Based on Rayleigh Model
2.2. Analytical Solution
3. Numerical Simulations and Discussion
3.1. Methodology of Numerical Simulations
3.2. Calculated Critical Frequencies
3.2.1. Results
3.2.2. Discussion
3.3. Calculated Unbalance Response
3.3.1. Results
3.3.2. Discussion
- (1)
- Accuracy
- (2)
- Numerical stability
3.4. Calculating Speed
4. Conclusions
- (1)
- The functional relationship between the unbalance response and location on the shaft, rotor unbalance (amplitude and angle), each bearing’s stiffness and damping coefficients and rotor’s inherent parameters is obtained based on CRDAM.
- (2)
- Numerical simulations indicate that the obtained unbalance response calculated by the three methods are almost equal when the frequency is away from the critical frequency, but they are different when the frequency is near the critical frequency. Moreover, CRDAM is closer to FEM than Ricatti. Although the critical frequencies calculated by them are almost equal to each other, the critical frequency calculated by CRDAM is closer to the critical frequency calculated by FEM than the critical frequency obtained by Ricatti. The unbalance response obtained by CRDAM is also closer to the unbalance response obtained by FEM than the unbalance response by Ricatti, especially when the frequency is smaller than the low order frequency.
- (3)
- Numerical simulations indicate that the calculating speed of CRDAM is the second fast among the three methods, but the speed of CRDAM becomes faster than Ricatti when the rotor is complicated. Simulations also show that the numerical stability of FEM and Ricatti is better than CRDAM. However, the numerical instability of CRDAM occurs when the frequency is high, often higher than the fifth order frequency.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix B
g1.1 | h1.1 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Order | CRDAM | FE | Ricatti | C/F(%) | R/F(%) | CRDAM | FEM | Ricatti | C/F(%) | R/F(%) |
1th | 33 | 33 | 33 | 0.000 | 0.000 | 33 | 33 | 33 | 0.000 | 0.000 |
2th | 209 | 209 | 209 | 0.000 | 0.000 | 212 | 212 | 212 | 0.000 | 0.000 |
3th | 355 | 352 | 360 | 0.852 | 2.273 | 598 | 597 | 596 | 0.168 | 0.168 |
g2.2 | h2.2 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Order | CRDAM (Hz) | FEM (Hz) | Ricatti (Hz) | C/F (%) | R/F (%) | CRDAM (Hz) | FEM (Hz) | Ricatti (Hz) | C/F (%) | R/F (%) |
1th | 16 | 16 | 16 | 0.000 | 0.000 | 16 | 16 | 16 | 0.000 | 0.000 |
2th | 104 | 104 | 104 | 0.000 | 0.000 | 104 | 104 | 104 | 0.000 | 0.000 |
3th | 294 | 294 | 294 | 0.000 | 0.000 | 295 | 295 | 295 | 0.000 | 0.000 |
4th | 568 | 568 | 571 | 0.000 | 0.528 | 570 | 570 | 573 | 0.000 | 0.526 |
5th | 774 | 773 | 777 | 0.129 | 0.517 | 799 | 798 | 799 | 0.125 | 0.125 |
g3.3 | h3.3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Order | CRDAM | FE | Ricatti | C/F(%) | R/F(%) | CRDAM | FEM | Ricatti | C/F(%) | R/F(%) |
1th | 14 | 14 | 14 | 0.000 | 0.000 | 92 | 92 | 92 | 0.000 | 0.000 |
2th | 91 | 91 | 91 | 0.000 | 0.000 | 258 | 258 | 260 | 0.000 | 0.775 |
3th | 256 | 256 | 258 | 0.000 | 0.781 | 433 | 433 | 436 | 0.000 | 0.693 |
4th | 390 | 390 | 392 | 0.000 | 0.513 | 458 | 457 | 458 | 0.219 | 0.219 |
5th | 431 | 431 | 434 | 0.000 | 0.696 | 562 | 562 | 567 | 0.000 | 0.890 |
6th | 549 | 549 | 555 | 0.000 | 1.093 | 892 | 892 | 911 | 0.000 | 2.130 |
7th | 928 | 916 | 962 | 1.310 | 5.022 | 947 | 934 | 982 | 1.392 | 5.139 |
g4.4 | h4.4 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Order | CRDAM | FE | Ricatti | C/F(%) | R/F(%) | CRDAM | FEM | Ricatti | C/F(%) | R/F(%) |
1th | 143 | 143 | 144 | 0.000 | 0.699 | 265 | 265 | 267 | 0.000 | 0.755 |
2th | 253 | 253 | 255 | 0.000 | 0.791 | 303 | 303 | 305 | 0.000 | 0.660 |
3th | 259 | 259 | 260 | 0.000 | 0.386 | 451 | 451 | 458 | 0.000 | 1.552 |
4th | 279 | 279 | 280 | 0.000 | 0.358 | 534 | 531 | 550 | 0.565 | 3.578 |
5th | 447 | 447 | 454 | 0.000 | 1.566 | / | / | / | / | / |
6th | 529 | 526 | 544 | 0.570 | 3.422 | / | / | / | / | / |
7th | 739 | 738 | 759 | 0.136 | 2.846 | / | / | / | / | / |
g5.5 | h5.5 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Order | CRDAM | FE | Ricatti | C/F(%) | R/F(%) | CRDAM | FEM | Ricatti | C/F(%) | R/F(%) |
1th | 171 | 171 | 172 | 0.000 | 0.585 | 122 | 122 | 123 | 0.000 | 0.820 |
2th | 341 | 340 | 350 | 0.294 | 2.941 | 192 | 192 | 193 | 0.000 | 0.521 |
3th | 484 | 483 | 497 | 0.207 | 2.899 | 199 | 199 | 200 | 0.000 | 0.503 |
4th | / | / | / | / | / | 206 | 206 | 207 | 0.000 | 0.485 |
5th | / | / | / | / | / | 243 | 241 | 242 | 0.830 | 0.415 |
6th | / | / | / | / | / | 429 | 428 | 442 | 0.234 | 3.271 |
g1.2 | h1.2 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Order | CRDAM (Hz) | FEM (Hz) | Ricatti (Hz) | C/F (%) | R/F (%) | CRDAM (Hz) | FEM (Hz) | Ricatti (Hz) | C/F (%) | R/F (%) |
1th | 22 | 22 | 22 | 0.000 | 0.000 | 22 | 22 | 22 | 0.000 | 0.000 |
2th | 87 | 87 | 87 | 0.000 | 0.000 | 87 | 87 | 87 | 0.000 | 0.000 |
3th | 198 | 198 | 199 | 0.000 | 0.505 | 198 | 198 | 199 | 0.000 | 0.505 |
4th | 329 | 329 | 330 | 0.000 | 0.304 | 329 | 329 | 330 | 0.000 | 0.304 |
5th | 517 | 517 | 520 | 0.000 | 0.580 | 518 | 517 | 521 | 0.193 | 0.774 |
6th | 749 | 747 | 757 | 0.268 | 1.339 | 753 | 750 | 749 | 0.400 | 0.133 |
7th | 952 | 948 | 963 | 0.422 | 1.582 | 965 | 960 | 977 | 0.521 | 1.771 |
g1.3 | h1.3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Order | CRDAM | FE | Ricatti | C/F(%) | R/F(%) | CRDAM | FEM | Ricatti | C/F(%) | R/F(%) |
1th | 10 | 10 | 10 | 0.000 | 0.000 | 10 | 10 | 10 | 0.000 | 0.000 |
2th | 39 | 39 | 40 | 0.000 | 2.564 | 39 | 39 | 40 | 0.000 | 2.564 |
3th | 86 | 86 | 86 | 0.000 | 0.000 | 86 | 86 | 86 | 0.000 | 0.000 |
4th | 151 | 151 | 152 | 0.000 | 0.662 | 151 | 151 | 152 | 0.000 | 0.662 |
5th | 225 | 225 | 226 | 0.000 | 0.444 | 225 | 225 | 226 | 0.000 | 0.444 |
6th | 335 | 334 | 338 | 0.299 | 1.198 | 335 | 335 | 338 | 0.000 | 0.896 |
7th | 425 | 424 | 429 | 0.236 | 1.179 | 426 | 425 | 431 | 0.235 | 1.412 |
8th | 526 | 525 | 531 | 0.190 | 1.143 | 529 | 529 | 535 | 0.000 | 1.134 |
9th | 686 | 683 | 700 | 0.439 | 2.489 | 691 | 688 | 705 | 0.436 | 2.471 |
10th | 847 | 843 | 868 | 0.474 | 2.966 | 853 | 848 | 874 | 0.590 | 3.066 |
g1.4 | h1.4 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Order | CRDAM | FE | Ricatti | C/F(%) | R/F(%) | CRDAM | FEM | Ricatti | C/F(%) | R/F(%) |
1th | 22 | 22 | 22 | 0.000 | 0.000 | 22 | 22 | 22 | 0.000 | 0.000 |
2th | 49 | 49 | 49 | 0.000 | 0.000 | 49 | 49 | 49 | 0.000 | 0.000 |
3th | 86 | 86 | 86 | 0.000 | 0.000 | 86 | 86 | 86 | 0.000 | 0.000 |
4th | 125 | 125 | 126 | 0.000 | 0.800 | 125 | 125 | 126 | 0.000 | 0.800 |
5th | 237 | 237 | 239 | 0.000 | 0.844 | 190 | 189 | 191 | 0.529 | 1.058 |
6th | 298 | 298 | 301 | 0.000 | 1.007 | 238 | 238 | 240 | 0.000 | 0.840 |
7th | 383 | 382 | 389 | 0.262 | 1.832 | 299 | 299 | 302 | 0.000 | 1.003 |
8th | 470 | 469 | 480 | 0.213 | 2.345 | 384 | 384 | 391 | 0.000 | 1.823 |
9th | / | / | / | / | / | 471 | 470 | 481 | 0.213 | 2.340 |
g1.5 | h1.5 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Order | CRDAM | FE | Ricatti | C/F(%) | R/F(%) | CRDAM | FEM | Ricatti | C/F(%) | R/F(%) |
1th | 4 | 4 | 4 | 0.000 | 0.000 | 4 | 4 | 4 | 0.000 | 0.000 |
2th | 14 | 14 | 14 | 0.000 | 0.000 | 14 | 14 | 14 | 0.000 | 0.000 |
3th | 32 | 32 | 32 | 0.000 | 0.000 | 32 | 32 | 32 | 0.000 | 0.000 |
4th | 81 | 81 | 82 | 0.000 | 1.235 | 55 | 55 | 55 | 0.000 | 0.000 |
5th | 120 | 120 | 121 | 0.000 | 0.833 | 81 | 81 | 82 | 0.000 | 1.235 |
6th | 153 | 153 | 154 | 0.000 | 0.654 | 120 | 120 | 121 | 0.000 | 0.833 |
7th | 193 | 193 | 194 | 0.000 | 0.518 | 153 | 153 | 154 | 0.000 | 0.654 |
8th | 245 | 245 | 249 | 0.000 | 1.633 | 193 | 193 | 195 | 0.000 | 1.036 |
9th | 301 | 300 | 307 | 0.333 | 2.333 | 246 | 246 | 250 | 0.000 | 1.626 |
10th | / | / | / | / | / | 301 | 301 | 308 | 0.000 | 2.326 |
Appendix C
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Symbols | Description |
---|---|
is the dimensionless lateral displacement of the shaft in y direction. | |
is the dimensionless lateral displacement of the shaft in x direction. | |
is the dimensionless quantityy of z. | |
is the dimensionless ; is the position of NO. i bearing; i = 1 − n. | |
uib is the dimensionless lateral displacement of the shaft in y direction at position zib. | |
is the dimensionless lateral displacement of the shaft in x direction at position . | |
is the dimensionless ; is the position of the number j disc; j = 1 − m. | |
is the dimensionless lateral displacement of the shaft in y direction at position . | |
is the dimensionless lateral displacement of the shaft in x direction at position . |
Parameter | Meaning |
---|---|
r_shaft | Radius of the shaft |
p_shaft | Density of the shaft |
E_shaft | Elastic modulus of the shaft |
L_shaft | Length of the shaft |
r_disc | Radius of the disc |
p_disc | Density of the disc |
E_disc | Elastic modulus of the disc |
L_disc | Width of the disc |
Parameter | Value | Parameter | Value |
---|---|---|---|
r_shaft | 10 × 10−3 m | L_shaft of h4.4, h1.4, g4.4, g1.4 | 1800 × 4 × 10−3 m |
p_shaft | 7800 kg·m−3 | L_shaft of h3.3, h1.3, g3.3, g1.3 | 1800 × 3 × 10−3 m |
E_shaft | 2.1 × 1011 Pa | L_shaft of h2.2, h1.2, g2.2, g1.2 | 1800 × 2 × 10−3 m |
L_shaft of h5.5, h1.5, g5.5, g1.5 | 1800 × 5 × 10−3 m | L_shaft of h1.1, g1.1 | 1800 × 1 × 10−3 m |
Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|
0.05 kg | 0.15 kg | 0.01 kg | |||
30 × 10−3 m | 10 × 10−3 m | 20 × 10−3 m | |||
45° | 90° | 20° | |||
Z1d | Point 8 | Z2d | Point 20 | Z3d | Point 29 |
0.10 kg | 0.05 kg | r_disc of #1~#5 disc | 50 × 10−3 m | ||
15 × 10−3 m | 40 × 10−3 m | p_disc of of #1~#5 disc | 7800 kg·m−3 | ||
190° | 290° | E_disc of of #1~#5 disc | 2.1 × 1011 Pa | ||
Z4d | Point 40 | Z5d | Point 55 | L_disc of of #1~#5 disc | 10 × 10−3 m |
Parameter | Value | Parameter | Value |
---|---|---|---|
N/m | |||
N·s/m | N·s/m | ||
N/m | |||
N·s/m | N·s/m | ||
N/m | |||
N·s/m | N·s/m | ||
N/m | |||
N·s/m | N·s/m | ||
N/m | |||
N·s/m | |||
Z1b | Point 2 | Z2b | Point 13 |
Z3b | Point 14 | Z4b | Point 24 |
Z5b | Point 25 | Z6b | Point 35 |
Z7b | Point 36 | Z8b | Point 46 |
Z9b | Point 47 | Z10b | Point 57 |
Z1b | Z2b | Z3b | Z4b | Z5b | Z6b | Z7b | Z8b |
---|---|---|---|---|---|---|---|
Point 2 | Point 13 | Point 14 | Point 24 | Point 25 | Point 35 | Point 36 | Point 46 |
Z9b | Z10b | , i = 1 to 10 | |||||
Point 47 | Point 57 | N/m | N·s/m |
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Wang, A.; Bi, Y.; Xia, Y.; Cheng, X.; Yang, J.; Meng, G. Continuous Rotor Dynamics of Multi-Disc and Multi-Span Rotor: A Theoretical and Numerical Investigation on the Continuous Model and Analytical Solution for Unbalance Responses. Appl. Sci. 2022, 12, 4351. https://doi.org/10.3390/app12094351
Wang A, Bi Y, Xia Y, Cheng X, Yang J, Meng G. Continuous Rotor Dynamics of Multi-Disc and Multi-Span Rotor: A Theoretical and Numerical Investigation on the Continuous Model and Analytical Solution for Unbalance Responses. Applied Sciences. 2022; 12(9):4351. https://doi.org/10.3390/app12094351
Chicago/Turabian StyleWang, Aiming, Yujie Bi, Yun Xia, Xiaohan Cheng, Jie Yang, and Guoying Meng. 2022. "Continuous Rotor Dynamics of Multi-Disc and Multi-Span Rotor: A Theoretical and Numerical Investigation on the Continuous Model and Analytical Solution for Unbalance Responses" Applied Sciences 12, no. 9: 4351. https://doi.org/10.3390/app12094351
APA StyleWang, A., Bi, Y., Xia, Y., Cheng, X., Yang, J., & Meng, G. (2022). Continuous Rotor Dynamics of Multi-Disc and Multi-Span Rotor: A Theoretical and Numerical Investigation on the Continuous Model and Analytical Solution for Unbalance Responses. Applied Sciences, 12(9), 4351. https://doi.org/10.3390/app12094351