New Numerical Results from Simulations of Beams and Space Frame Systems with a Tuned Mass Damper
Abstract
:1. Introduction
2. Geometry and Theoretical Formulations
The Space Frame with a TMD
3. Artificial Neural Networks
4. Numerical Results and Discussions
4.1. Numerical Results for Free Vibration Analysis of the Beam and Frame System with a TMD
4.1.1. Accuracy Studies
4.1.2. Numerical Results for Free Vibration Analysis of the Plane Beam with a TMD
The Beam with One TMD
The Beam with Three TMDs
4.2. Numerical Results for Free Vibration Analysis of the Space Frame System with a TMD
4.3. Setup of the ANN Model
4.4. Numerical Results for Vibration Analysis of Beam with a TMD under Random Loading
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
Appendix A.1. The Shape Functions
Appendix A.2. The Distributed Mass Matrix Components
Appendix A.3. The Element Matrices
Appendix A.4. The Relationship of the Nodal Displacement Components between the Local Coordinate System and the Global Coordinate System of the Structure
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mT/m0 (%) | cT (Ns/m) | kT (N/m) | Beam without TMD | Beam with One TMD | ||||||
---|---|---|---|---|---|---|---|---|---|---|
f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | |||
1 | 100 | 2000 | 1.161 | 5.093 | 19.933 | 63.312 | 0.887 | 1.178 | 5.093 | 19.937 |
2 | 0.631 | 1.171 | 5.093 | 19.937 | ||||||
3 | 0.516 | 1.169 | 5.093 | 19.937 | ||||||
4 | 0.447 | 1.169 | 5.093 | 19.937 | ||||||
5 | 0.400 | 1.169 | 5.093 | 19.937 | ||||||
6 | 0.365 | 1.169 | 5.093 | 19.937 | ||||||
7 | 0.338 | 1.168 | 5.093 | 19.937 | ||||||
8 | 0.316 | 1.168 | 5.093 | 19.937 | ||||||
9 | 0.298 | 1.168 | 5.093 | 19.937 | ||||||
10 | 0.283 | 1.168 | 5.093 | 19.937 |
mT/m0 (%) | cT (Ns/m) | kT (N/m) | Beam without TMD | Beam with One TMD | ||||||
---|---|---|---|---|---|---|---|---|---|---|
f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | |||
10 | 10 | 2000 | 1.161 | 5.093 | 19.933 | 63.312 | 0.283 | 1.168 | 5.093 | 19.937 |
20 | 0.283 | 1.168 | 5.093 | 19.937 | ||||||
100 | 0.283 | 1.168 | 5.093 | 19.937 | ||||||
150 | 0.283 | 1.168 | 5.093 | 19.937 | ||||||
300 | 0.283 | 1.168 | 5.093 | 19.937 | ||||||
500 | 0.283 | 1.168 | 5.093 | 19.937 | ||||||
700 | 0.283 | 1.168 | 5.093 | 19.937 | ||||||
800 | 0.283 | 1.168 | 5.093 | 19.937 | ||||||
1000 | 0.283 | 1.168 | 5.093 | 19.937 |
mT/m0 (%) | cT (Ns/m) | kT (N/m) | Beam without TMD | Beam with One TMD | ||||||
---|---|---|---|---|---|---|---|---|---|---|
f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | |||
10 | 100 | 100 | 1.161 | 5.093 | 19.933 | 63.312 | 0.063 | 1.161 | 5.093 | 19.935 |
500 | 0.142 | 1.162 | 5.093 | 19.936 | ||||||
1000 | 0.200 | 1.164 | 5.093 | 19.936 | ||||||
1500 | 0.245 | 1.166 | 5.093 | 19.936 | ||||||
2000 | 0.283 | 1.168 | 5.093 | 19.937 | ||||||
3000 | 0.345 | 1.172 | 5.093 | 19.937 | ||||||
5000 | 0.442 | 1.181 | 5.093 | 19.938 | ||||||
8000 | 0.552 | 1.197 | 5.093 | 19.940 | ||||||
10,000 | 0.611 | 1.209 | 5.093 | 19.941 |
mT2/m0 (%) | mT1/m0 (%) | cT1 (Ns/m) | kT1 (N/m) | Beam without TMD | Beam with Three TMDs | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | ||||
1 | 4.0 | 102 | 2000 | 1.161 | 5.093 | 19.933 | 63.312 | 0.447 | 0.450 | 0.888 | 1.185 |
1 | 3.0 | 0.516 | 0.519 | 0.888 | 1.186 | ||||||
1 | 2.0 | 0.631 | 0.638 | 0.888 | 1.187 | ||||||
1 | 1.0 | 0.876 | 0.900 | 0.901 | 1.193 | ||||||
2 | 4.0 | 0.447 | 0.450 | 0.631 | 1.179 | ||||||
2 | 3.0 | 0.512 | 0.519 | 0.631 | 1.179 | ||||||
2 | 2.0 | 0.626 | 0.636 | 0.637 | 1.180 | ||||||
2 | 1.0 | 0.631 | 0.888 | 0.900 | 1.187 | ||||||
3 | 3 | 0.512 | 0.519 | 0.520 | 1.178 | ||||||
3 | 2 | 0.516 | 0.632 | 0.636 | 1.179 | ||||||
3 | 1 | 0.516 | 0.888 | 0.900 | 1.186 | ||||||
4 | 3 | 0.447 | 0.516 | 0.519 | 1.178 | ||||||
4 | 2 | 0.447 | 0.631 | 0.636 | 1.179 | ||||||
4 | 1 | 0.447 | 0.888 | 0.900 | 1.185 | ||||||
5 | 2 | 0.400 | 0.631 | 0.636 | 1.178 | ||||||
5 | 1 | 0.400 | 0.888 | 0.900 | 1.185 |
mT2/m0 (%) | cT1 (Ns/m) | kT1 (N/m) | Beam without TMD | Beam with Three TMDs | ||||||
---|---|---|---|---|---|---|---|---|---|---|
f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | |||
3 | 10 | 2000 | 1.161 | 5.093 | 19.933 | 63.312 | 0.512 | 0.519 | 0.520 | 1.178 |
20 | 0.512 | 0.519 | 0.520 | 1.178 | ||||||
100 | 0.512 | 0.519 | 0.520 | 1.178 | ||||||
150 | 0.512 | 0.519 | 0.520 | 1.178 | ||||||
300 | 0.512 | 0.519 | 0.520 | 1.178 | ||||||
500 | 0.512 | 0.519 | 0.520 | 1.178 | ||||||
700 | 0.512 | 0.519 | 0.520 | 1.178 | ||||||
800 | 0.512 | 0.519 | 0.520 | 1.178 | ||||||
1000 | 0.512 | 0.519 | 0.520 | 1.178 |
mT2/m0 (%) | cT1 (Ns/m) | kT1 (N/m) | Beam without TMD | Beam with Three TMDs | ||||||
---|---|---|---|---|---|---|---|---|---|---|
f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | |||
3 | 100 | 100 | 1.161 | 5.093 | 19.933 | 63.312 | 0.116 | 0.117 | 0.118 | 1.161 |
500 | 0.259 | 0.260 | 0.261 | 1.164 | ||||||
1000 | 0.365 | 0.367 | 0.368 | 1.168 | ||||||
1500 | 0.445 | 0.450 | 0.451 | 1.173 | ||||||
2000 | 0.512 | 0.519 | 0.520 | 1.178 | ||||||
3000 | 0.621 | 0.636 | 0.637 | 1.190 | ||||||
5000 | 0.780 | 0.821 | 0.822 | 1.224 | ||||||
8000 | 0.924 | 1.037 | 1.040 | 1.306 | ||||||
10,000 | 0.976 | 1.159 | 1.162 | 1.382 |
Method | |||
---|---|---|---|
SAP-2000 | 4.437 | 6.375 | 6.784 |
This work | 4.426 | 6.039 | 6.522 |
mT/m0 (%) | cT (Ns/m) | kT (N/m) | The Frame System without TMD | The Frame System with One TMD | ||||||
---|---|---|---|---|---|---|---|---|---|---|
f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | |||
1 | 103 | 104 | 0.704 | 0.961 | 1.038 | 1.283 | 0.475 | 0.704 | 0.961 | 1.038 |
2 | 0.336 | 0.704 | 0.961 | 1.038 | ||||||
3 | 0.274 | 0.704 | 0.961 | 1.038 | ||||||
4 | 0.237 | 0.704 | 0.961 | 1.038 | ||||||
5 | 0.212 | 0.704 | 0.961 | 1.038 | ||||||
6 | 0.194 | 0.704 | 0.961 | 1.038 | ||||||
7 | 0.179 | 0.704 | 0.961 | 1.038 | ||||||
8 | 0.168 | 0.704 | 0.961 | 1.038 | ||||||
9 | 0.158 | 0.704 | 0.961 | 1.038 | ||||||
10 | 0.150 | 0.704 | 0.961 | 1.038 |
mT/m0 (%) | cT (Ns/m) | kT (N/m) | The Frame System without TMD | The Frame System with One TMD | ||||||
---|---|---|---|---|---|---|---|---|---|---|
f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | |||
10 | 103 | 5000 | 0.704 | 0.961 | 1.038 | 1.283 | 0.106 | 0.704 | 0.961 | 1.038 |
7000 | 0.126 | 0.704 | 0.961 | 1.038 | ||||||
8000 | 0.134 | 0.704 | 0.961 | 1.038 | ||||||
10,000 | 0.150 | 0.704 | 0.961 | 1.038 | ||||||
12,000 | 0.164 | 0.704 | 0.961 | 1.038 | ||||||
15,000 | 0.184 | 0.704 | 0.961 | 1.038 | ||||||
18,000 | 0.201 | 0.704 | 0.961 | 1.038 | ||||||
20,000 | 0.212 | 0.704 | 0.961 | 1.038 | ||||||
25,000 | 0.237 | 0.704 | 0.961 | 1.038 |
mT/m0 (%) | cT (Ns/m) | kT (N/m) | The Frame System without TMD | The Frame System with One TMD | ||||||
---|---|---|---|---|---|---|---|---|---|---|
f1 (Hz) | f2 (Hz) | f1 (Hz) | f2 (Hz) | f1 (Hz) | f2 (Hz) | f1 (Hz) | f2 (Hz) | |||
10 | 100 | 104 | 0.704 | 0.961 | 1.038 | 1.283 | 0.150 | 0.704 | 0.961 | 1.038 |
500 | 0.150 | 0.704 | 0.961 | 1.038 | ||||||
800 | 0.150 | 0.704 | 0.961 | 1.038 | ||||||
1000 | 0.150 | 0.704 | 0.961 | 1.038 | ||||||
1400 | 0.150 | 0.704 | 0.961 | 1.038 | ||||||
1600 | 0.150 | 0.704 | 0.961 | 1.038 | ||||||
1800 | 0.150 | 0.704 | 0.961 | 1.038 | ||||||
2000 | 0.150 | 0.704 | 0.961 | 1.038 | ||||||
2500 | 0.150 | 0.704 | 0.961 | 1.038 |
Order | mT2/m0 (%) | mT1/m0 (%) | cT (Ns/m) | kT (N/m) | Target f1 (Hz) | Predict f1 (Hz) |
---|---|---|---|---|---|---|
1 | 1 | 4.0 | 100 | 2000 | 0.447 | 0.447 |
2 | 1 | 3.0 | 100 | 2000 | 0.516 | 0.516 |
3 | 1 | 2.0 | 100 | 2000 | 0.631 | 0.631 |
4 | 1 | 1.0 | 100 | 2000 | 0.876 | 0.876 |
5 | 2 | 4.0 | 100 | 2000 | 0.447 | 0.447 |
6 | 2 | 3.0 | 100 | 2000 | 0.512 | 0.512 |
7 | 2 | 2.0 | 100 | 2000 | 0.626 | 0.626 |
8 | 2 | 1.0 | 100 | 2000 | 0.631 | 0.631 |
9 | 3 | 3 | 100 | 2000 | 0.512 | 0.512 |
10 | 3 | 2 | 100 | 2000 | 0.516 | 0.516 |
11 | 3 | 1 | 100 | 2000 | 0.516 | 0.5199 |
12 | 4 | 3 | 100 | 2000 | 0.447 | 0.447 |
13 | 4 | 2 | 100 | 2000 | 0.447 | 0.447 |
14 | 4 | 1 | 100 | 2000 | 0.447 | 0.447 |
15 | 5 | 2 | 100 | 2000 | 0.400 | 0.400 |
16 | 5 | 1 | 100 | 2000 | 0.400 | 0.400 |
17 | 3 | 3 | 10 | 2000 | 0.512 | 0.512 |
18 | 3 | 3 | 20 | 2000 | 0.512 | 0.512 |
19 | 3 | 3 | 150 | 2000 | 0.512 | 0.512 |
20 | 3 | 3 | 300 | 2000 | 0.512 | 0.512 |
21 | 3 | 3 | 500 | 2000 | 0.512 | 0.512 |
22 | 3 | 3 | 700 | 2000 | 0.512 | 0.512 |
23 | 3 | 3 | 800 | 2000 | 0.512 | 0.5097 |
24 | 3 | 3 | 1000 | 2000 | 0.512 | 0.512 |
25 | 3 | 3 | 100 | 100 | 0.116 | 0.116 |
26 | 3 | 3 | 100 | 500 | 0.259 | 0.259 |
27 | 3 | 3 | 100 | 1000 | 0.365 | 0.365 |
28 | 3 | 3 | 100 | 1500 | 0.445 | 0.445 |
29 | 3 | 3 | 100 | 3000 | 0.621 | 0.6195 |
30 | 3 | 3 | 100 | 5000 | 0.780 | 0.790 |
31 | 3 | 3 | 100 | 8000 | 0.924 | 0.924 |
32 | 3 | 3 | 100 | 10,000 | 0.976 | 0.976 |
EAverage (%) | 0.0017 | |||||
Emax (%) | 0.7484 |
Order | mT/m0 (%) | cT (Ns/m) | kT (N/m) | Target f1 (Hz) | Predict f2 (Hz) |
---|---|---|---|---|---|
1 | 1 | 1000 | 10,000 | 0.475 | 0.475 |
2 | 2 | 1000 | 10,000 | 0.336 | 0.336 |
3 | 3 | 1000 | 10,000 | 0.274 | 0.2739 |
4 | 4 | 1000 | 10,000 | 0.237 | 0.2372 |
5 | 5 | 1000 | 10,000 | 0.212 | 0.2121 |
6 | 6 | 1000 | 10,000 | 0.194 | 0.1936 |
7 | 7 | 1000 | 10,000 | 0.179 | 0.1794 |
8 | 8 | 1000 | 10,000 | 0.168 | 0.1678 |
9 | 9 | 1000 | 10,000 | 0.158 | 0.158 |
10 | 10 | 1000 | 10,000 | 0.150 | 0.150 |
11 | 10 | 1000 | 5000 | 0.106 | 0.1061 |
12 | 10 | 1000 | 7000 | 0.126 | 0.1256 |
13 | 10 | 1000 | 8000 | 0.134 | 0.1343 |
14 | 10 | 1000 | 12,000 | 0.164 | 0.1643 |
15 | 10 | 1000 | 15,000 | 0.184 | 0.1838 |
16 | 10 | 1000 | 18,000 | 0.201 | 0.2013 |
17 | 10 | 1000 | 20,000 | 0.212 | 0.2118 |
18 | 10 | 1000 | 25,000 | 0.237 | 0.237 |
19 | 10 | 100 | 10,000 | 0.150 | 0.150 |
20 | 10 | 500 | 10,000 | 0.150 | 0.150 |
21 | 10 | 800 | 10,000 | 0.150 | 0.150 |
22 | 10 | 1400 | 10,000 | 0.150 | 0.150 |
23 | 10 | 1600 | 10,000 | 0.150 | 0.150 |
24 | 10 | 1800 | 10,000 | 0.150 | 0.150 |
25 | 10 | 2000 | 10,000 | 0.150 | 0.150 |
26 | 10 | 2500 | 10,000 | 0.150 | 0.150 |
EAverage (%) | 0.0073 | ||||
Emax (%) | 0.3052 |
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Tho, N.C.; Ta, N.T.; Thom, D.V. New Numerical Results from Simulations of Beams and Space Frame Systems with a Tuned Mass Damper. Materials 2019, 12, 1329. https://doi.org/10.3390/ma12081329
Tho NC, Ta NT, Thom DV. New Numerical Results from Simulations of Beams and Space Frame Systems with a Tuned Mass Damper. Materials. 2019; 12(8):1329. https://doi.org/10.3390/ma12081329
Chicago/Turabian StyleTho, Nguyen Chi, Nguyen Tri Ta, and Do Van Thom. 2019. "New Numerical Results from Simulations of Beams and Space Frame Systems with a Tuned Mass Damper" Materials 12, no. 8: 1329. https://doi.org/10.3390/ma12081329
APA StyleTho, N. C., Ta, N. T., & Thom, D. V. (2019). New Numerical Results from Simulations of Beams and Space Frame Systems with a Tuned Mass Damper. Materials, 12(8), 1329. https://doi.org/10.3390/ma12081329