Dynamic Finite Element Model Updating Based on Correlated Mode Auto-Pairing and Adaptive Evolution Screening
Abstract
:1. Introduction
- (1)
- The correlation between analytical and experimental modes can be uniquely established by frequencies or/and MAC thresholds, denoted as case 1↔1 in this paper.
- (2)
- Similarly, there is a one-to-many or many-to-one correlation between analytical and experimental modes established by frequencies or/and MAC thresholds. This type of situation is denoted as case 1↔n.
- (3)
- Compared with measuring data, one or several analytical modes are missing. Alternatively, one or several experimental modes are missing compared with numerical results. This type of situation is denoted as case 0↔n.
2. Basic Dynamic Model Updating by Pairing Correlated Modes
3. Correlated Mode Auto-Pairing and Adaptive Evolution Screening Method
3.1. Correlated Mode Auto-Pairing Strategy
3.2. Population Evolution Screening Mechanism Provided by an Evolutionary Algorithm
3.3. Dynamic Model Updating Evaluation Criteria
4. Dynamic Model Updating Examples
4.1. Proof Examples: Dynamic Model Updating of a Thin Plate
4.1.1. Updating Based on the Simulated “Experimental” Data
4.1.2. Updating Based on the Impact Modal Test Data
4.2. Supplementary Example: Dynamic Model Updating of the F-Shaped Structure
4.3. Engineering Example: Dynamic Model Updating of an Intermediate Case
5. Conclusions
- (1)
- To solve the problem that modal data cannot be fully exploited when natural frequencies and mode shapes of complex structures change dramatically, the correlated mode auto-pairing strategy is proposed. The one-to-one CMPs are determined adaptively by the auto-pairing strategy before each iteration based on the one-to-one symbiotic relationship between natural frequencies and mode shapes. This strategy, liberating from dependence on demanding pre-analysis, is suitable for all the 1↔1, 1↔n, and 0↔n cases in dynamic FE model updating.
- (2)
- To further screen the CMPs determined by the auto-pairing strategy in each generation and search the global minimum, the population evolution mechanism is used to simultaneously urge the updating parameters and the CMPs towards the ideal results. Combined with the auto-pairing strategy, the adaptive switch from penalized to correlated subitem can screen the CMPs further during iteration to ensure that all the potential of analytical or experimental modes can be fully exploited.
- (3)
- The examples of a thin plate with non-uniform thickness and small holes validated the accuracy and effectiveness of the proposed method. In the updating of the IMC with different cross-sectional shapes of hollow struts and multi-layer thin-walled complex structure, all the radial modes within the range of four times the operating speed were auto-screened out and updated well. The updated results show the great potential of the proposed CMPES method for complex engineering problems.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Model Type | Area Type | Non-Dimensional Ratio of Density | Non-Dimensional Ratio of Modulus of Elasticity | Non-Dimensional Ratio of Poisson’s Ratio |
---|---|---|---|---|
Initial FE | Hole | 0.5 | 1.2 | 0.75 |
Main | 1.0 | 1.0 | 1.0 | |
‘Experimental’ | Hole | 0.89 | 1.05 | 0.97 |
Main | 0.64 | 0.88 | 1.07 |
Mode No. | “Experimental”Frequency (Hz) | Initial Analytical Modal Analysis | |||
---|---|---|---|---|---|
Frequency (Hz) | FRE (%) | MAC | 1-MAC | ||
1 | 161.13 | 227.18 | 41.00 | 1.00 | 0.00 |
2 | 209.56 | 303.10 | 44.64 | 1.00 | 0.00 |
3 | 445.72 | 624.76 | 40.17 | 1.00 | 0.00 |
4 | 455.18 | 658.21 | 44.61 | 1.00 | 0.00 |
5 | 732.77 | 1016.49 | 38.72 | 0.98 | 0.02 |
6 | 780.83 | 1124.19 | 43.97 | 1.00 | 0.00 |
7 | 855.55 | 1175.48 | 37.40 | 0.99 | 0.01 |
8 | 892.67 | 1222.67 | 36.97 | 0.97 | 0.03 |
9 | 1154.68 | 1629.08 | 41.08 | 0.99 | 0.01 |
10 | 1219.74 | 1742.04 | 42.82 | 1.00 | 0.00 |
RMSE | 41.23 | 0.0126 | |||
CV | 0.0107 |
Area | Boundary | Non-Dimensional Ratio of Density | Non-Dimensional Ratio of Modulus of Elasticity | Non-Dimensional Ratio of Poisson’s Ratio |
Main | Lower | 0.1 | 0.1 | 0.1 |
Upper | 1.5 | 1.5 | 1.2 | |
Hole | Lower | 0.1 | 0.1 | 0.1 |
Upper | 1.5 | 1.5 | 1.3 |
Parameter Name | Value | Parameter Name | Value |
---|---|---|---|
Number of individuals | 50 | Migration direction | forward |
Number of elite | 5 | Migration Interval | 15 |
Crossover probability | 0.75 | Migration probability | 0.1 |
Maximum number of generations | 200 | Function Tolerance | 10-6 |
Mode No. | MUUM | CMPES | ||||||
---|---|---|---|---|---|---|---|---|
Frequency (Hz) | FRE (%) | MAC | 1-MAC | Frequency (Hz) | FRE (%) | MAC | 1-MAC | |
1 | 162.45 | 0.82 | 1.00 | 0.00 | 161.13 | 0.01 | 1.00 | 0.00 |
2 | 211.12 | 0.75 | 1.00 | 0.00 | 209.55 | −0.01 | 1.00 | 0.00 |
3 | 449.25 | 0.79 | 1.00 | 0.00 | 445.72 | 0.00 | 1.00 | 0.00 |
4 | 458.70 | 0.77 | 1.00 | 0.00 | 455.08 | −0.02 | 1.00 | 0.00 |
5 | 738.63 | 0.80 | 1.00 | 0.00 | 732.60 | −0.02 | 1.00 | 0.00 |
6 | 787.00 | 0.79 | 1.00 | 0.00 | 780.86 | 0.00 | 1.00 | 0.00 |
7 | 861.69 | 0.72 | 1.00 | 0.00 | 855.35 | −0.02 | 1.00 | 0.00 |
8 | 899.26 | 0.74 | 1.00 | 0.00 | 892.64 | 0.00 | 1.00 | 0.00 |
9 | 1163.85 | 0.79 | 1.00 | 0.00 | 1154.77 | 0.01 | 1.00 | 0.00 |
10 | 1229.49 | 0.80 | 1.00 | 0.00 | 1219.92 | 0.01 | 1.00 | 0.00 |
RMSE | 0.78 | 0.01 | ||||||
CV | 0.00 | 0.00 |
Mode No. | Experimental Frequency (Hz) | Modal Damping Ratios (%) | Initial Analytical Modal Analysis | |||
---|---|---|---|---|---|---|
Frequency (Hz) | FRE (%) | MAC | 1-MAC | |||
1 | 145.56 | 0.34 | 227.18 | 56.08 | 0.9549 | 0.0451 |
2 | 193.65 | 0.24 | 303.10 | 56.52 | 0.9728 | 0.0272 |
3 | 397.43 | 0.30 | 624.76 | 57.20 | 0.7745 | 0.2255 |
4 | 403.57 | 0.40 | 658.21 | 63.10 | 0.9286 | 0.0714 |
5 | 627.03 | 0.32 | 1016.49 | 62.11 | 0.9918 | 0.0082 |
6 | 695.33 | 0.50 | 1124.19 | 61.68 | 0.7069 | 0.2931 |
7 | 718.01 | 0.55 | 1175.48 | 63.71 | 0.9631 | 0.0369 |
8 | 773.24 | 0.41 | 1222.67 | 58.12 | 0.6200 | 0.3800 |
9 | 976.84 | 0.62 | 1629.08 | 66.77 | 0.9492 | 0.0508 |
10 | 1073.56 | 0.54 | 1742.04 | 62.27 | 0.6504 | 0.3496 |
RMSE | 60.85 | 0.2038 | ||||
CV | 0.1724 |
Mode No. | MUUM | CMPES | ||||||
---|---|---|---|---|---|---|---|---|
Frequency (Hz) | FRE (%) | MAC | 1-MAC | Frequency (Hz) | FRE (%) | MAC | 1-MAC | |
1 | 141.47 | −2.81 | 0.9489 | 0.0511 | 145.72 | 0.11 | 0.9537 | 0.0463 |
2 | 195.02 | 0.70 | 0.9725 | 0.0275 | 183.65 | −5.16 | 0.9716 | 0.0284 |
3 | 383.20 | −3.58 | 0.7507 | 0.2493 | 393.05 | −1.10 | 0.7951 | 0.2049 |
4 | 413.39 | 2.43 | 0.9301 | 0.0699 | 396.65 | −1.71 | 0.9276 | 0.0724 |
5 | 614.27 | −2.04 | 0.9917 | 0.0083 | 632.10 | 0.81 | 0.9842 | 0.0158 |
6 | 702.44 | 1.02 | 0.7191 | 0.2809 | 692.68 | −0.38 | 0.7244 | 0.2756 |
7 | 722.52 | 0.63 | 0.9723 | 0.0277 | 719.50 | 0.21 | 0.9562 | 0.0438 |
8 | 746.54 | −3.45 | 0.6106 | 0.3894 | 766.81 | −0.83 | 0.6559 | 0.3441 |
9 | 1019.28 | 4.34 | 0.9588 | 0.0412 | 1029.38 | 5.38 | 0.9437 | 0.0563 |
10 | 1087.17 | 1.27 | 0.6394 | 0.3606 | 1094.62 | 1.96 | 0.6657 | 0.3343 |
RMSE | 2.55 | 0.2082 | 2.55 | 0.1902 | ||||
CV | 0.1784 | 0.1552 |
Group Name | Correlated Mode Pairs | Uncorrelated Mode Pairs |
---|---|---|
MUUM | (A1, E1) (A1, E2) (A3, E3) (A4, E4) (A5, E5) (A6, E6) (A7, E7) (A8, E8) (A9, E9) (A10, E10) | (A7, E6) (A6, E7) |
MUUM-I | (A1, E1) (A1, E2) (A3, E3) (A4, E4) (A5, E5) (A6, E6) (A7, E7) (A8, E8) (A9, E9) (A11, E10) | (A7, E6) (A8, E6) (A6, E7) (A8, E7) (A6, E8) (A7, E8) |
MUUM-II | (A1, E1) (A1, E2) (A3, E3) (A4, E4) (A5, E5) (A6, E6) (A7, E7) (A8, E8) (A9, E9) | (A7, E6) (A8, E6) (A6, E7) (A8, E7) (A6, E8) (A7, E8) |
Group Name | RMSEFRE (%) | RMSEMAC | CV | Group Name | RMSEFRE (%) | RMSEMAC | CV |
---|---|---|---|---|---|---|---|
MUUM | 2.55 | 0.2082 | 0.1784 | CMPES | 2.55 | 0.1902 | 0.1552 |
MUUM-I | 6.36 | 0.2147 | 0.1852 | CMPES-I | 2.55 | 0.1902 | 0.1552 |
MUUM-II | 7.46 | 0.2161 | 0.1870 | CMPES-II | 2.54 | 0.1903 | 0.1552 |
Mode No. | “Experimental”Frequency (Hz) | MUUM | CMPES | ||||||
---|---|---|---|---|---|---|---|---|---|
Frequency (Hz) | FRE (%) | MAC | 1-MAC | Frequency (Hz) | FRE (%) | MAC | 1-MAC | ||
1 | 12.01 | 12.17 | 1.34 | 1.00 | 0.00 | 12.01 | 0.00 | 1.00 | 0.00 |
2 | 54.73 | 54.86 | 0.24 | 1.00 | 0.00 | 54.73 | 0.00 | 1.00 | 0.00 |
3 | 62.38 | 62.52 | 0.21 | 1.00 | 0.00 | 62.38 | 0.00 | 1.00 | 0.00 |
4 | 260.71 | 260.85 | 0.05 | 1.00 | 0.00 | 260.71 | 0.00 | 1.00 | 0.00 |
5 | 1110.40 | 1110.46 | 0.01 | 1.00 | 0.00 | 1110.41 | 0.00 | 1.00 | 0.00 |
6 | 1150.50 | 1150.57 | 0.01 | 1.00 | 0.00 | 1150.52 | 0.00 | 1.00 | 0.00 |
7 | 1208.40 | 1208.67 | 0.02 | 1.00 | 0.00 | 1208.37 | 0.00 | 1.00 | 0.00 |
8 | 2130.40 | 2130.63 | 0.01 | 1.00 | 0.00 | 2130.46 | 0.00 | 1.00 | 0.00 |
9 | 2245.80 | 2245.80 | 0.00 | 1.00 | 0.00 | 2245.77 | 0.00 | 1.00 | 0.00 |
RMSE | 0.46 | 0.00 | |||||||
CV | 0.00 | 0.00 |
Mode No. | Mode Description | Experimental Frequency (Hz) | Updated Analytical Frequency (Hz) | FRE (%) | MAC | 1-MAC |
---|---|---|---|---|---|---|
1 | Radial | 248.89 | 270.87 | 8.83 | 0.8752 | 0.1248 |
2 | Radial | 297.38 | 272.17 | −8.48 | 0.8752 | 0.1248 |
3 | Radial | 327.13 | 339.00 | 3.63 | 0.9420 | 0.0580 |
4 | Radial | 345.59 | 341.18 | −1.28 | 0.8616 | 0.1384 |
5 | Torsional | 389.10 | ||||
6 | Radial | 587.44 | 644.32 | 9.68 | 0.7409 | 0.2591 |
7 | Radial | 647.03 | 649.33 | 0.36 | 0.8444 | 0.1556 |
8 | Radial | 689.50 | 761.62 | 10.46 | 0.5987 | 0.4013 |
9 | Axial | 739.64 | ||||
10 | Radial | 746.37 | 760.85 | 1.94 | 0.6187 | 0.3813 |
11 | Axial | 774.78 | ||||
12 | Axial | 821.28 | ||||
13 | Local | 831.38 | ||||
14 | Radial | 843.40 | 867.74 | 2.89 | 0.6091 | 0.3909 |
RMSE | 6.50 | 0.2591 | ||||
CV | 0.1736 |
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Shao, H.; Zuo, Y.; Jiang, Z. Dynamic Finite Element Model Updating Based on Correlated Mode Auto-Pairing and Adaptive Evolution Screening. Appl. Sci. 2022, 12, 3175. https://doi.org/10.3390/app12063175
Shao H, Zuo Y, Jiang Z. Dynamic Finite Element Model Updating Based on Correlated Mode Auto-Pairing and Adaptive Evolution Screening. Applied Sciences. 2022; 12(6):3175. https://doi.org/10.3390/app12063175
Chicago/Turabian StyleShao, Huajin, Yanfei Zuo, and Zhinong Jiang. 2022. "Dynamic Finite Element Model Updating Based on Correlated Mode Auto-Pairing and Adaptive Evolution Screening" Applied Sciences 12, no. 6: 3175. https://doi.org/10.3390/app12063175
APA StyleShao, H., Zuo, Y., & Jiang, Z. (2022). Dynamic Finite Element Model Updating Based on Correlated Mode Auto-Pairing and Adaptive Evolution Screening. Applied Sciences, 12(6), 3175. https://doi.org/10.3390/app12063175