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Natural Frequencies
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n f (Hz) n f (Hz) # elements n f (Hz) # elements1.00 3.61 1.00 1.1069 10 1.00 3.612 102.00 22.64 2.00 23.6568 10 2.00 22.637 103.00 63.37 3.00 65.2272 10 3.00 63.387 104.00 124.21 4.00 133.169 10 4.00 124.231 105.00 205.32 5.00 206.266 10 5.00 205.432 106.00 306.72 6.00 304.67 10 6.00 307.076 10
n f (Hz) n f (Hz) # elements n f (Hz) # elements1.00 3.61 1.00 3.49432 10.00 1.00 3.52 10.002.00 22.64 2.00 19.049 10.00 2.00 19.237 10.003.00 63.37 3.00 45.7123 10.00 3.00 77.67 10.004.00 124.21 4.00 135.206 10.00 4.00 145.58 10.005.00 205.32 5.00 211.95 10.00 5.00 211.89 10.006.00 306.72 6.00 305.29 10.00 6.00 303.07 10.00
n f (Hz) n f (Hz) # elements n f (Hz) # elements1.00 3.61 1.00 3.50034 10.00 1.00 1.526 10.002.00 22.64 2.00 18.9704 10.00 2.00 19.65 10.003.00 63.37 3.00 45.27 10.00 3.00 66.87 10.004.00 124.21 4.00 133.11 10.00 4.00 134.087 10.005.00 205.32 5.00 210 10.00 5.00 203.24 10.006.00 306.72 6.00 324 10.00 6.00 315.85 10.00
1 2 3 4 5 6
Analytical Natural Frequencies - Cantiliver Beam
Natural Frequencies - ABAQUS-2D Wire
Natural Frequencies - Comsol-2D Wire
Analytical Natural Frequencies - Cantiliver Beam
Natural Frequencies - ABAQUS-3D Solid
Natural Frequencies - Comsol-3D Solid
Analytical Natural Frequencies - Cantiliver Beam
Natural Frequencies - ABAQUS-3D Beam
Natural Frequencies - Comsol-3D Beam
1 2 3 4 5 6
3.61
22.6
4
63.3
7
124.
21
205.
32
306.
72
3.61
2
22.6
37
63.3
87
124.
231
205.
432
307.
076
1.10
69
23.6
568
65.2
272
133.
169
206.
266
304.
67
1.52
6
19.6
5
66.8
7
134.
087
203.
24
315.
85
3.50
034
18.9
704
45.2
7
133.
11
210
324
3.52
19.2
37
77.6
7
145.
58
211.
89
303.
07
3.49
432
19.0
49
45.7
123
135.
206
211.
95
305.
29
Analytical Comsol 2D Wire Abaqus 2D Wire Comsol 3D Beam Abaqus 3D Beam Comsol 3D Solid Abaqus 3D Solid
10 3.61 22.64 63.37 124.21 205.32 306.7220 3.61 22.64 63.37 124.21 205.32 306.7250 3.61 22.64 63.37 124.21 205.32 306.72
1 2 3 4 5 6
3.61
22.6
4
63.3
7
124.
21
205.
32
306.
72
3.61
2
22.6
37
63.3
87
124.
231
205.
432
307.
076
1.10
69
23.6
568
65.2
272
133.
169
206.
266
304.
67
1.52
6
19.6
5
66.8
7
134.
087
203.
24
315.
85
3.50
034
18.9
704
45.2
7
133.
11
210
324
3.52
19.2
37
77.6
7
145.
58
211.
89
303.
07
3.49
432
19.0
49
45.7
123
135.
206
211.
95
305.
29
Analytical Comsol 2D Wire Abaqus 2D Wire Comsol 3D Beam Abaqus 3D Beam Comsol 3D Solid Abaqus 3D Solid
2D Wire Linear -Abaqus 2D Wire Quadratic -Abaqus
10 20 50 10 20 501 1.1069 1 1.11003 1 1.11091 1 1.11108 1 1.11108 1 1.111082 5.99898 2 6.04471 2 6.05767 2 6.06061 2 6.06017 2 6.060143 8.82975 3 8.83656 3 8.83847 3 8.83884 3 8.83883 3 8.838834 14.3176 4 14.5026 4 14.555 4 14.5723 4 14.5655 4 14.5655 23.6568 5 24.1891 5 24.3402 5 24.408 5 24.3716 5 24.36916 26.2718 6 26.4552 6 26.5067 6 26.5174 6 26.5166 6 26.51657 33.1593 7 34.4209 7 34.778 7 34.9757 7 34.8547 7 34.84648 42.1119 8 43.9108 8 44.1488 8 44.2055 8 44.1949 8 44.19429 43.067 9 44.6207 9 45.322 9 45.7752 9 45.4773 9 45.4562
10 50.0937 10 54.4227 10 55.5713 10 56.3893 10 55.8293 10 55.78811 56.5841 11 60.9021 11 61.578 11 61.8843 11 61.7163 11 61.702112 58.8018 12 61.0956 12 61.7472 12 61.931 12 61.8757 12 61.871913 60.3315 13 64.9863 13 66.5442 13 67.5646 13 66.8848 13 66.83114 61.8324 14 68.7568 14 70.1473 14 71.0829 14 70.4706 14 70.422515 65.2272 15 75.178 15 77.6608 15 79.1633 15 78.1879 15 78.103716 66.835 16 77.9037 16 79.2848 16 79.7497 16 79.563 16 79.549917 73.0886 17 78.6243 17 81.2723 17 83.395 17 81.9233 17 81.801218 74.1263 18 85.6833 18 89.6617 18 91.8487 18 90.4876 18 90.353619 81.7586 19 88.7274 19 92.9268 19 96.566 19 94.0207 19 93.780520 85.5757 20 94.2315 20 96.7441 20 97.7493 20 97.2634 20 97.228121 89.2989 21 95.7221 21 102.223 21 104.474 21 103.591 21 103.34322 95.9557 22 98.9362 22 104.748 22 112.484 22 106.299 22 105.92823 96.3451 23 104.789 23 114.108 23 116.047 23 114.987 23 114.90724 102.458 24 108.697 24 114.338 24 120.796 24 116.823 24 116.20525 103.973 25 109.978 25 117.45 25 127.325 25 119.222 25 118.86326 107.294 26 113.752 26 125.682 26 134.772 26 129.332 26 128.26427 109.43 27 116.633 27 130.935 27 135.91 27 132.748 27 132.587
28 110.628 28 121.773 28 131.359 28 144.065 28 133.457 28 132.99129 112.193 29 124.354 29 137.043 29 153.843 29 141.654 29 140.20430 112.326 30 125.047 30 143.614 30 153.981 30 Cantiliver 30 147.089
Cantilivern w (rad/s) f (Hz)
1.00 1.88 22.70 3.612.00 4.69 142.23 22.643.00 7.85 398.18 63.374.00 11.00 780.42 124.215.00 14.14 1290.09 205.326.00 17.28 1927.17 306.727.00 20.42 2691.66 428.398.00 23.56 3583.58 570.349.00 26.70 4602.90 732.58
10.00 29.85 5749.65 915.0911 32.98672 7023.809 1117.87412 36.12832 8425.386 1340.94213 39.26991 9954.379 1584.28914 42.4115 11610.79 1847.91415 45.55309 13394.61 2131.81916 48.69469 15305.85 2436.00217 51.83628 17344.51 2760.46418 54.97787 19510.58 3105.20619 58.11946 21804.07 3470.22620 61.26106 24224.98 3855.52521 64.40265 26773.3 4261.10222 67.54424 29449.03 4686.95923 70.68583 32252.19 5133.09524 73.82743 35182.76 5599.50925 76.96902 38240.74 6086.20326 80.11061 41426.14 6593.17527 83.25221 44738.96 7120.42628 86.3938 48179.19 7667.957
29 89.53539 51746.84 8235.76630 92.67698 55441.91 8823.853
3D Beam Linear -Abaqus 3D Beam Quadratic-Abaqus
10 20 50 10 20 501 3.50034 1 3.51023 1 3.51301 1 3.51354 1 3.51354 1 3.513542 3.50034 2 3.51023 2 3.51301 2 3.51354 2 3.51354 2 3.513543 16.9244 3 16.9374 3 16.9411 3 16.9418 3 16.9418 3 16.94184 18.9704 4 19.1151 4 19.156 4 19.1653 4 19.1639 4 19.16395 18.9704 5 19.1151 5 19.156 5 19.1653 5 19.1639 5 19.16396 27.9221 6 27.9437 6 27.9497 6 27.9509 6 27.9508 6 27.95087 45.2763 7 45.8611 7 46.027 7 46.0817 7 46.0601 7 46.05878 45.2763 8 45.8611 8 46.027 8 46.0817 8 46.0601 8 46.05879 50.3564 9 50.7079 9 50.8066 9 50.8271 9 50.8255 9 50.8254
10 74.8094 10 76.4928 10 76.9706 10 77.1848 10 77.0698 10 77.06211 74.8094 11 76.4928 11 76.9706 11 77.1848 11 77.0698 11 77.06212 82.5485 12 83.6587 12 83.8215 12 83.8554 12 83.8527 12 83.852613 83.0788 13 84.1657 13 84.6219 13 84.7307 13 84.7104 13 84.70914 104.859 14 108.848 14 109.978 14 110.603 14 110.22 14 110.19415 104.859 15 108.848 15 109.978 15 110.603 15 110.22 15 110.19416 112.708 16 117.105 16 118.354 16 118.706 16 118.6 16 118.59317 133.169 17 138.858 17 139.611 17 139.79 17 139.757 17 139.75418 133.169 18 141.103 18 143.321 18 144.754 18 143.812 18 143.74519 136.19 19 141.103 19 143.321 19 144.754 19 143.812 19 143.74520 140.092 20 149.322 20 151.969 20 152.86 20 152.502 20 152.47721 158.41 21 172.1 21 175.732 21 178.319 21 176.548 21 176.41722 158.41 22 172.1 22 175.732 22 178.319 22 176.548 22 176.41723 164.027 23 180.618 23 185.434 23 187.361 23 186.429 23 186.36224 178.935 24 192.589 24 194.727 24 195.695 24 195.164 24 195.11925 178.935 25 192.589 25 194.727 25 195.695 25 195.164 25 195.11926 183.923 26 193.201 26 195.262 26 195.843 26 195.668 26 195.65627 185.947 27 205.505 27 210.431 27 213.658 27 211.508 27 211.338
28 190.785 28 205.505 28 210.431 28 213.658 28 211.508 28 211.33829 190.785 29 210.801 29 218.716 29 222.433 29 220.401 29 220.24830 195.531 30 217.428 30 221.825 30 224.784 30 222.848 30 222.696
3D Solid Linear -Abaqus 3D Solid Quadratic-Abaqus
10 20 50 10 20 501 3.49432 1 3.51368 1 3.5207 1 3.520312 3.49432 2 3.51368 2 3.5207 2 3.520313 16.8074 3 16.9332 3 16.9815 3 16.97684 19.049 4 19.1619 4 19.2067 4 19.20165 19.049 5 19.1619 5 19.2067 5 19.20166 22.7851 6 27.978 6 27.9868 6 27.98587 22.7851 7 45.5702 7 46.2169 7 46.1838 22.7851 8 45.5702 8 46.2169 8 46.1839 22.7851 9 45.5702 9 50.9364 9 50.9201
10 22.7851 10 45.5702 10 77.4818 10 77.339811 22.7851 11 45.5702 11 77.4818 11 77.339812 22.7851 12 45.5702 12 83.839 12 83.833513 22.7851 13 45.5702 13 84.8816 13 84.833914 22.7851 14 45.5702 14 111.1 14 110.66715 22.7851 15 45.5702 15 111.1 15 110.66716 26.9329 16 45.5702 16 118.854 16 118.70117 26.9329 17 45.5702 17 139.245 17 139.20818 27.9561 18 45.5702 18 145.403 18 144.3819 31.5939 19 45.5702 19 145.403 19 144.3820 36.2195 20 45.5702 20 152.946 20 152.50721 36.264 21 45.5702 21 178.744 21 176.91722 42.0549 22 45.5702 22 178.744 22 176.91723 42.0549 23 45.5702 23 187.314 23 186.24224 45.7123 24 45.5702 24 193.054 24 192.58825 45.7123 25 45.5702 25 193.054 25 192.58826 46.9398 26 45.5702 26 193.394 26 193.23127 46.9398 27 46.0576 27 194.981 27 195.24
28 48.427 28 46.0576 28 211.954 28 209.97529 48.427 29 46.1392 29 211.954 29 209.97530 49.9932 30 46.1392 30 222.174 30 219.901
2D Wire Linear -COMSOL 3D Beam Linear -COMSOL
10 20 50 10 20 501 3.612 1 3.612 1 3.612 1 1.526 1 1.526 1 1.5262 22.637 2 22.637 2 22.637 2 1.526 2 1.526 2 1.5263 27.963 3 22.963 3 22.963 3 3.56 3 3.56 3 3.564 63.38 4 63.387 4 63.387 4 7.11 4 7.11 4 7.115 84.19 5 84.197 5 84.197 5 7.11 5 7.11 5 7.116 124.23 6 124.231 6 124.231 6 7.25 6 7.25 6 7.257 141.355 7 141.355 7 141.355 7 7.45 7 7.45 7 7.458 200.06 8 200.06 8 200.06 8 7.45 8 7.45 8 7.459 205.43 9 205.432 9 205.432 9 8.62 9 8.62 9 8.62
10 260.93 10 260.939 10 260.939 10 8.62 10 8.62 10 8.6211 307.07 11 307.076 11 307.076 11 9.87 11 9.87 11 9.8712 324.6 12 324.6 12 324.6 12 18.93 12 18.93 12 18.9313 391.59 13 391.595 13 391.595 13 19.43 13 19.43 13 19.4314 429.351 14 429.351 14 429.351 14 19.65 14 19.65 14 19.6515 462.3 15 462.303 15 462.303 15 19.65 15 19.65 15 19.6516 536.75 16 536.759 16 536.759 16 19.96 16 19.96 16 19.9617 572.56 17 572.57 17 572.57 17 20.081 17 20.081 17 20.08118 614.33 18 614.433 18 614.433 18 20.691 18 20.691 18 20.69119 737.19 19 614.334 19 614.334 19 20.691 19 20.691 19 20.69120 769.95 20 693.246 20 693.246 20 23.83 20 23.83 20 23.8321 838.68 21 737.193 21 737.193 21 26.34 21 26.34 21 26.3422 891.66 22 769.95 22 769.95 22 26.34 22 26.34 22 26.3423 920.82 23 838.68 23 838.68 23 28.84 23 28.84 23 28.8424 923.86 24 891.663 24 891.663 24 29.47 24 29.47 24 29.4725 1133.391 25 920.82 25 920.82 25 29.47 25 29.47 25 29.4726 1366.639 26 923.864 26 923.864 26 30.257 26 30.257 26 30.25727 1624.051 27 1133.39 27 1133.39 27 30.612 27 30.612 27 30.612
28 1903.29 28 1366.639 28 1366.639 28 35.911 28 35.911 28 35.91129 2181.228 29 1624.051 29 1624.051 29 39.275 29 39.275 29 39.275
30 1903.291 30 1903.291 30 40.529 30 40.529 30 40.529
3D SOLID Linear -COMSOL
10 20 501 3.523 1 3.523 1 3.5232 3.523 2 3.523 2 3.5233 17.1 3 17.1 3 17.14 19.235 4 19.235 4 19.2355 19.237 5 19.237 5 19.2376 27.991 6 27.991 6 27.9917 46.312 7 46.312 7 46.3128 46.322 8 46.322 8 46.3229 51.303 9 51.303 9 51.303
10 77.675 10 77.675 10 77.67511 77.695 11 77.695 11 77.69512 83.856 12 83.856 12 83.85613 85.615 13 85.615 13 85.61514 111.344 14 111.344 14 111.34415 111.38 15 111.38 15 111.3816 120.141 16 120.141 16 120.14117 139.249 17 139.249 17 139.24918 145.58 18 145.58 18 145.5819 145.645 19 145.645 19 145.64520 154.857 20 154.857 20 154.85721 178.734 21 178.734 21 178.73422 178.79 22 178.79 22 178.7923 189.91 23 189.91 23 189.9124 193.216 24 193.216 24 193.21625 193.226 25 193.226 25 193.22626 193.33 26 193.33 26 193.3327 197.373 27 197.373 27 197.373
28 211.969 28 211.969 28 211.96929 1600.754 29 1600.754 29 1600.75430 3131.816 30 3131.816 30 3131.816
2D Wire Linear -Abaqus
Cantiliver beam Analytical Natural Frequencies, Euler-Bernoulli 10 20n knl wn f(Hz) 1 0.0284 1
E 30000 Pa 1 1.875 0.179 0.0286 2 0.1748 2h 0.05 m 2 4.696 1.125 0.1791 3 0.1751 3b 0.1 m 3 7.855 3.149 0.5011 4 0.4685 4L 1 m 4 10.996 6.170 0.9820 5 0.4794 5
h/L 0.05 5 14.137 10.199 1.6232 6 0.8830 6Ro 2400 kg/m3 6 17.279 15.236 2.4248 7 0.8891 7I 1.0417E-06 m4 7 20.420 21.279 3.3867 8 0.9157 8
P2 0.05103 8 23.562 28.331 4.5090 9 1.4189 99 26.704 36.389 5.7915 10 1.4684 1010 29.845 45.455 7.2344 11 1.7605 11
12 2.0399 1213 2.1154 1314 2.6272 1415 2.7282 1516 2.8250 1617 3.4449 1718 3.4777 1819 3.5465 1920 4.1209 2021 4.1988 2122 4.3067 2223 4.6373 2324 4.6666 2425 5.1092 2526 5.8802 2627 6.6149 2728 7.30886 2829 7.95775 29
30 8.55757 3031323334353637
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
Thin Beam -10 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
Thin Beam -10 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
9.0000
Thin Beam -20 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
9.0000
Thin Beam -20 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
2D Wire Linear -Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus
20 50 10 20 50 10 200.0285 1 0.0285 1 0.0285 1 0.0285 1 0.0285 1 0.0284 1 0.02850.1763 2 0.1767 2 0.1768 2 0.1768 2 0.1768 2 0.0565 2 0.05670.1789 3 0.1800 3 0.1802 3 0.1802 3 0.1802 3 0.1748 3 0.17630.4838 4 0.4866 4 0.4872 4 0.4869 4 0.4869 4 0.3376 4 0.34040.4850 5 0.4884 5 0.4896 5 0.4893 5 0.4892 5 0.4297 5 0.43000.8837 6 0.8838 6 0.8839 6 0.8839 6 0.8839 6 0.4794 6 0.48500.9257 7 0.9318 7 0.9350 7 0.9327 7 0.9325 7 0.8830 7 0.88370.9281 8 0.9370 8 0.9417 8 0.9394 8 0.9392 8 0.8882 8 0.89891.4863 9 1.4979 9 1.5096 9 1.5000 9 1.4993 9 0.9157 9 0.92811.4908 10 1.5081 10 1.5229 10 1.5131 10 1.5124 10 1.2784 10 1.28741.7660 11 1.7675 11 1.7678 11 1.7678 11 1.7678 11 1.4684 11 1.49082.1471 12 2.1674 12 2.2009 12 2.1719 12 2.1699 12 1.6099 12 1.63732.1547 13 2.1843 13 2.2233 13 2.1938 13 2.1917 13 2.0957 13 2.13682.6455 14 2.6507 14 2.6517 14 2.6517 14 2.6517 14 2.1154 14 2.15472.8897 15 2.9233 15 3.0036 15 2.9329 15 2.9278 15 2.4365 15 2.49882.9007 16 2.9484 16 3.0377 16 2.9656 16 2.9603 16 2.6272 16 2.64553.5210 17 3.5332 17 3.5359 17 3.5356 17 3.5355 17 2.8250 17 2.90073.6954 18 3.7500 18 3.9186 18 3.7690 18 3.7581 18 2.8614 18 2.97303.7102 19 3.7846 19 3.9673 19 3.8142 19 3.8028 19 3.3036 19 3.43544.3911 20 4.4149 20 4.4206 20 4.4195 20 4.4194 20 3.5465 20 3.71024.5468 21 4.6340 21 4.9562 21 4.6689 21 4.6478 21 3.5566 21 3.79094.5652 22 4.6788 22 5.0230 22 4.7280 22 4.7060 22 4.1518 22 4.39115.2544 23 5.2955 23 5.3061 23 5.3035 23 5.3033 23 4.1643 23 4.41185.4267 24 5.5633 24 6.1763 24 5.6235 24 5.5862 24 4.1988 24 4.56525.4484 25 5.6185 25 6.1931 25 5.6974 25 5.6583 25 4.3067 25 4.58556.1096 26 6.1747 26 6.2872 26 6.1876 26 6.1872 26 4.6666 26 5.35176.3188 27 6.5278 27 7.0824 27 6.6260 27 6.5641 27 4.6694 27 5.40116.3431 28 6.5934 28 7.1837 28 6.7153 28 6.6505 28 4.9221 28 5.44846.9553 29 7.0525 29 7.2975 29 7.0718 29 7.0711 29 5.0595 29 6.0850
7.2074 30 7.5189 30 7.9750 30 7.6714 30 7.5742 30 5.3250 30 6.10967.2331 31 7.5946 31 8.7999 31 7.7765 31 7.6746 31 5.4595 31 6.34317.7904 32 7.9285 32 8.8720 32 7.9563 32 7.9550 32 5.5512 32 6.38268.0765 33 8.5294 33 8.9154 33 8.7565 33 8.6106 33 5.8802 33 6.78088.1023 34 8.6143 34 8.8411 34 8.7245 34 5.9706 34 7.23318.6134 35 8.8025 35 8.8776 35 8.8389 35 7.3089 35 7.33918.9100 36 8.5576 36 7.43478.9345 37 7.7904
38 8.042839 8.102340 8.256441 8.601342 8.9345
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
Thin Beam -10 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
Thin Beam -50 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
Thin Beam -10 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
9.0000
Thin Beam -20 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
Thin Beam -50 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
9.0000
Thin Beam -20 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
3D Beam Linear -Abaqus 3D Beam Quadratic -Abaqus 3D SOLID Linear -Abaqus
50 10 20 50 10 20 301 0.0285 1 0.0285 1 0.0285 1 0.0285 1 0.0283 1 0.0285 12 0.1767 2 0.1768 2 0.1768 2 0.1768 2 0.0563 2 0.0566 23 0.1800 3 0.1802 3 0.1802 3 0.1802 3 0.1757 3 0.1766 34 0.4866 4 0.4872 4 0.4869 4 0.4869 4 0.3390 4 0.3409 45 0.4884 5 0.4896 5 0.4893 5 0.4892 5 0.4654 5 0.4436 56 0.8838 6 0.8839 6 0.8839 6 0.8839 6 0.4879 6 0.4874 67 0.9318 7 0.9350 7 0.9327 7 0.9325 7 0.7205 7 0.8841 78 0.9370 8 0.9417 8 0.9394 8 0.9392 8 0.7205 8 0.9025 89 1.4979 9 1.5096 9 1.5000 9 1.4993 9 0.7205 9 0.9168 910 1.5081 10 1.5229 10 1.5131 10 1.5124 10 0.7205 10 1.3292 1011 1.7675 11 1.7678 11 1.7678 11 1.7678 11 0.7205 11 1.4411 1112 2.1674 12 2.2009 12 2.1719 12 2.1699 12 0.7205 12 1.4411 1213 2.1843 13 2.2233 13 2.1938 13 2.1917 13 0.7205 13 1.4411 1314 2.6507 14 2.6517 14 2.6517 14 2.6517 14 0.7205 14 1.4411 1415 2.9233 15 3.0036 15 2.9329 15 2.9278 15 0.7205 15 1.4411 1516 2.9484 16 3.0377 16 2.9656 16 2.9603 16 0.7205 16 1.4411 1617 3.5332 17 3.5359 17 3.5356 17 3.5355 17 0.8834 17 1.4411 1718 3.7500 18 3.9186 18 3.7690 18 3.7581 18 0.9008 18 1.4411 1819 3.7846 19 3.9673 19 3.8142 19 3.8028 19 0.9496 19 1.4411 1920 4.4149 20 4.4206 20 4.4195 20 4.4194 20 1.1610 20 1.4411 2021 4.6340 21 4.9562 21 4.6689 21 4.6478 21 1.3861 21 1.4411 2122 4.6788 22 5.0230 22 4.7280 22 4.7060 22 1.5627 22 1.4411 2223 5.2955 23 5.3061 23 5.3035 23 5.3033 23 1.6526 23 1.4411 2324 5.5633 24 6.1763 24 5.6235 24 5.5862 24 1.9356 24 1.4411 2425 5.6185 25 6.1931 25 5.6974 25 5.6583 25 1.9585 25 1.4411 2526 6.1747 26 6.2872 26 6.1876 26 6.1872 26 2.1540 26 1.4411 2627 6.5278 27 7.0824 27 6.6260 27 6.5641 27 2.2765 27 1.4411 2728 6.5934 28 7.1837 28 6.7153 28 6.6505 28 2.3295 28 1.4411 2829 7.0525 29 7.2975 29 7.0718 29 7.0711 29 2.5338 29 1.4411 29
30 7.5189 30 7.9750 30 7.6714 30 7.5742 30 2.6278 30 1.4411 3031 7.5946 31 8.7999 31 7.7765 31 7.6746 31 2.6384 31 1.5118 3132 7.9285 32 8.8720 32 7.9563 32 7.9550 32 2.7317 32 1.5124 3233 8.5294 33 8.9154 33 8.7565 33 8.6106 33 2.8030 33 1.6485 3334 8.6143 34 8.8411 34 8.7245 34 3.1176 34 1.7050 3435 8.8025 35 8.8776 35 8.8389 35 3.2455 35 1.7066 35
36 3.2522 36 1.7706 3637 3.4792 37 1.9484 3738 3.4818 38 1.9809 3839 3.6668 39 1.9873 3940 3.7630 40 2.1984 4041 3.8668 41 2.2084 4142 3.8879 42 2.2101 4243 4.0345 43 2.3044 4344 4.1449 44 2.3236 4445 4.3049 45 2.3793 4546 4.3144 46 2.5193 4647 4.4189 47 2.5231 4748 4.4217 48 2.5381 4849 4.4394 49 2.6462 4950 4.4993 50 2.6513 5051 4.5496 51 2.6900 5152 4.7028 52 2.7781 5253 4.8507 53 2.8575 5354 4.9206 54 2.9778 5455 4.9508 55 3.0060 5556 4.9967 56 3.0744 5657 5.0637 57 3.0778 5758 5.1230 58 3.0831 5859 5.2677 59 3.0852 5960 5.2692 60 3.0933 6061 5.2735 61 3.1300 61
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
Thin Beam -50 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
62 5.2991 62 3.2064 6263 5.3256 63 3.2822 6364 5.3505 64 3.3583 6465 5.3751 65 3.3768 6566 5.3914 66 3.4054 6667 5.4075 67 3.4735 6768 5.4192 68 3.4784 6869 5.4252 69 3.5167 6970 5.4663 70 3.5549 7071 5.4836 71 3.7009 7172 5.5350 72 3.7539 7273 5.5398 73 3.8123 7374 5.6414 74 3.8336 7475 5.7246 75 3.8769 7576 5.7733 76 3.8947 7677 5.7816 77 3.9288 7778 5.8725 78 3.9447 7879 5.9164 79 4.0285 7980 6.0307 80 4.0835 8081 6.0310 81 4.1054 8182 6.0600 82 4.1074 8283 6.0644 83 4.1487 8384 6.0644 84 4.2198 8485 6.1068 85 4.2799 8586 6.1418 86 4.3169 8687 6.1841 87 4.3246 8788 6.3239 88 4.3367 8889 6.3483 89 4.3900 8990 6.3847 90 4.4563 9091 6.5200 91 4.4782 9192 6.5919 92 4.5086 9293 6.6597 93 4.5254 93
1 2 3 4 5 6 7 8 9 100.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
Thin Beam -50 Elements
Analytical Natural Frequencies-Euler-Bernoulli 2D Wire Linear-Abaqus 2D Wire Quadratic -Abaqus 3D Beam Linear -Abaqus3D Beam Quadratic - Abaqus 3D Solid Linear - Abaqus 3D Solid Quadratic-Abaqus
94 6.7302 94 4.6222 9495 6.7487 95 4.6320 9596 6.7596 96 4.6832 9697 6.8524 97 4.7358 9798 6.8683 98 4.7487 9899 6.8853 99 4.7521 99
100 6.9462 100 4.7663 100101 6.9701 101 4.7913 101102 6.9774 102 4.7933 102103 7.1041 103 4.8078 103104 7.1158 104 4.8783 104105 7.1242 105 4.8820 105106 7.2053 106 4.8985 106107 7.2053 107 4.9861 107108 7.2053 108 5.0637 108109 7.2053 109 5.0760 109110 7.2053 110 5.0832 110111 7.2053 111 5.1128 111112 7.2053 112 5.1137 112113 7.2053 113 5.1531 113114 7.2053 114 5.2469 114115 7.2053 115 5.2665 115116 7.2343 116 5.2907 116117 7.2867 117 5.3155 117118 7.2976 118 5.3688 118119 7.3117 119 5.4101 119120 7.3427 120 5.4773 120121 7.3672 121 5.4804 121122 7.4087 122 5.4858 122123 7.4270 123 5.4869 123124 7.4716 124 5.4947 124125 7.5090 125 5.5312 125
126 7.5415 126 5.5734 126127 7.5679 127 5.6057 127128 7.5766 128 5.6185 128129 7.6025 129 5.6239 129130 7.6027 130 5.6440 130131 7.6217 131 5.6511 131132 7.6368 132 5.6817 132133 7.6900 133 5.7046 133134 7.7133 134 5.7062 134135 7.7278 135 5.7296 135136 7.8246 136 5.7653 136137 7.8400 137 5.7828 137138 7.8920 138 5.7938 138139 7.9890 139 5.8033 139140 8.0341 140 5.8464 140141 8.0348 141 5.8602 141142 8.0575 142 5.8604 142143 8.1041 143 5.8690 143144 8.1219 144 5.8823 144145 8.1289 145 5.9246 145146 8.1296 146 5.9480 146147 8.1692 147 5.9508 147148 8.1846 148 5.9767 148149 8.2403 149 5.9900 149150 8.2492 150 6.0027 150151 8.2795 151 6.1028 151152 8.2868 152 6.1367 152153 8.2913 153 6.1548 153154 8.2925 154 6.1632 154155 8.3331 155 6.2192 155156 8.3656 156 6.2476 156157 8.3764 157 6.2683 157
158 8.3908 158 6.2734 158159 8.4344 159 6.3024 159160 8.4453 160 6.3164 160161 8.4821 161 6.3284 161162 8.5043 162 6.4204 162163 8.5335 163 6.4626 163164 8.5430 164 6.4636 164165 8.5632 165 6.4834 165166 8.5645 166 6.4881 166167 8.6191 167 6.5060 167168 8.7407 168 6.5314 168169 8.7459 169 6.5529 169170 8.7547 170 6.5621 170171 8.7680 171 6.5753 171172 8.7883 172 6.6274 172173 8.8068 173 6.6546 173174 8.8265 174 6.6888 174175 8.8520 175 6.7019 175176 8.8617 176 6.7040 176177 8.8936 177 6.7168 177178 8.9116 178 6.7382 178179 8.9120 179 6.7798 179180 8.9201 180 6.8168 180181 8.9251 181 6.8258 181182 8.9519 182 6.8572 182183 8.9657 183 6.9082 183184 8.9864 184 6.9262 184
185 6.9853 185186 7.0404 186187 7.0567 187188 7.1382 188189 7.1580 189
190 7.1866 190191 7.1889 191192 7.1925 192193 7.2395 193194 7.2527 194195 7.2696 195196 7.2767 196197 7.3512 197198 7.4125 198199 7.4916 199200 7.5356 200201 7.5515 201202 7.5648 202203 7.5990 203204 7.6004 204205 7.6224 205206 7.6384 206207 7.6555 207208 7.6735 208209 7.6806 209210 7.6835 210211 7.6914 211212 7.7379 212213 7.7705 213214 7.8888 214215 7.9167 215216 7.9511 216217 7.9568 217218 7.9747 218219 8.0265 219220 8.0340 220221 8.0855 221
222 8.1299 222223 8.1426 223224 8.1628 224225 8.1881 225226 8.1901 226227 8.2686 227228 8.2833 228229 8.3358 229230 8.3580 230231 8.3584 231232 8.3601 232233 8.3794 233234 8.3884 234235 8.4320 235236 8.4320 236237 8.5031 237238 8.5333 238239 8.5948 239240 8.5972 240241 8.6069 241242 8.6165 242243 8.6259 243244 8.6599 244245 8.6681 245246 8.6708 246247 8.7145 247248 8.7157 248249 8.7622 249250 8.8172 250251 8.8330 251252 8.9365 252253 8.9551 253
254 8.9684 254255 8.9762 255256 8.9859 256
257258259260261262263264265266267268269270271272273274275276277278279280281282283284285
286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317
318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349
350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381
382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413
3D SOLID Linear -Abaqus 3D Solid Quadratic-Abaqus 2D Wire Linear -COMSOL
30 10 20 30 10 20 500.0285 1 0.0285 1 0.0285 1 0.0285 1 0.0571 1 0.0571 1 0.05710.0567 2 0.0567 2 0.0567 2 0.0567 2 0.357 2 0.3579 2 0.35790.1768 3 0.1770 3 0.1770 3 0.1769 3 1.0022 3 1.0022 3 1.00220.3412 4 0.3416 4 0.3415 4 0.3415 4 1.9642 4 1.9642 4 1.96420.4394 5 0.4360 5 0.4359 5 0.4359 5 3.2481 5 3.2481 5 3.24810.4874 6 0.4878 6 0.4874 6 0.4874 6 4.8552 6 4.8552 6 4.85520.8842 7 0.8843 7 0.8843 7 0.8843 7 6.7886 7 6.7886 7 6.78860.9028 8 0.9039 8 0.9033 8 0.90320.9350 9 0.9363 9 0.9339 9 0.93371.3180 10 1.3092 10 1.3089 10 1.30891.5061 11 1.5121 11 1.5022 11 1.50171.6484 12 1.6525 12 1.6487 12 1.64852.1616 13 2.1862 13 2.1758 13 2.17402.1616 14 2.2051 14 2.1852 14 2.18512.1616 15 2.5385 15 2.5248 15 2.52402.1616 16 2.6525 16 2.6524 16 2.65242.1616 17 3.0104 17 2.9388 17 2.93462.1616 18 3.0708 18 3.0669 18 3.06662.1616 19 3.5264 19 3.4893 19 3.48702.1616 20 3.9291 20 3.7776 20 3.76852.1616 21 3.9681 21 3.9561 21 3.95542.1616 22 4.4197 22 4.4186 22 4.41852.1616 23 4.5972 23 4.5138 23 4.50882.1616 24 4.8851 24 4.6808 24 4.66332.1616 25 4.9720 25 4.8546 25 4.85282.1616 26 5.7442 26 5.5795 26 5.56952.1616 27 5.8307 27 5.6391 27 5.60832.1616 28 6.1870 28 5.7642 28 5.76032.1616 29 6.2019 29 6.1815 29 6.1812
2.1616 30 6.8162 30 6.6459 30 6.59502.1616 31 6.9700 31 6.6737 31 6.65562.1616 32 7.2106 32 6.6866 32 6.67882.1616 33 7.8546 33 7.6237 33 7.60932.1616 34 7.9576 34 7.6960 34 7.61622.1616 35 8.3010 35 7.7870 35 7.75662.1616 36 8.8422 36 7.9391 36 7.93802.1616 37 8.9650 37 8.5770 37 8.55232.1616 38 8.7861 38 8.66652.1616 39 8.9116 39 8.86372.16162.16162.16162.18302.18382.19642.24582.24602.26832.33372.34582.34692.43792.47772.48122.50602.52332.57502.57872.63512.64292.6496
2.69442.73862.81222.82772.82932.83102.84672.85602.92232.93932.95132.97453.00363.02833.02993.07203.07423.12073.20473.23223.24653.24693.27943.32893.36693.37543.41173.45233.45443.47413.48243.4834
3.52843.54293.62153.68323.68783.71043.72773.72953.73133.74733.75943.79233.83133.85733.92053.93063.93973.95103.95243.96913.97574.02644.03264.03904.16074.18934.19934.20394.22474.23814.24074.2427
4.24324.26304.33924.36544.39844.40064.40584.44074.44934.47544.49374.52834.53754.56404.59754.60134.61454.63164.63774.69204.70104.74694.75024.75944.76494.79664.80144.81434.82514.82604.83264.8456
4.86994.88654.95484.98304.98785.00805.02465.03255.03665.09545.10895.13625.14545.15105.15965.17805.20725.21135.25725.29605.30335.31275.32255.32545.40145.41195.44395.46165.46195.47965.48555.5203
5.52675.53395.56175.56765.57235.61675.63675.65825.68815.69805.71165.71735.75005.75105.76555.77805.79075.80935.81995.84615.85345.86215.86235.86725.87625.87995.88505.90255.93525.94445.95565.9646
5.96995.98226.01756.05896.06566.06876.07526.07766.09066.09986.12036.12696.13416.14156.14506.14646.17376.19546.20536.20976.23466.24286.24756.25766.26726.27656.29666.30326.30666.36156.36856.3801
6.39686.41466.44716.44756.45176.45976.46286.51116.53246.56066.56376.56566.58566.60976.61426.61436.61736.62746.66396.69436.69806.70346.72856.74496.74526.75876.80226.81426.82866.84516.87766.8844
6.93906.94316.94396.95316.95776.96216.96776.97796.98106.98437.00117.00687.00897.07187.07497.10107.12627.13137.17367.23697.25537.27047.27417.29347.31857.36837.41407.41447.42687.44427.45027.4602
7.46237.48567.51227.51577.57667.57817.59187.61337.61347.62767.63397.63507.65937.69037.69477.69917.69957.76087.76177.78137.78887.79367.82167.83537.86337.86907.87467.89167.90457.91507.92317.9266
7.93777.96207.97397.98277.98797.98958.00978.01888.02878.06238.07308.07678.10928.14418.18698.20098.20168.21028.23438.26698.27958.28678.28948.30228.30588.35498.35808.37368.38358.38488.39418.4112
8.50038.50068.51308.56838.56888.60068.62428.66788.67068.67588.70528.72308.72318.74488.74978.75208.75598.76068.76568.76598.77758.82278.83878.85798.88158.90278.91158.94978.97178.97268.97298.9938
3D BEAM Linear -COMSOL 3D SOLID Linear -COMSOL
10 20 50 10 20 501 0.052 1 0.052 1 0.052 1 0.0286 1 0.0286 1 0.02862 0.7915 2 0.7915 2 0.7915 2 0.1784 2 0.1784 2 0.17843 1.4276 3 1.4276 3 1.4276 3 0.4958 3 0.4958 3 0.49584 1.8021 4 1.8021 4 1.8021 4 1.5744 4 1.5744 4 1.57445 2.5719 5 2.5719 5 2.5719 5 2.3248 5 2.3248 5 2.32486 3.349 6 3.349 6 3.349 6 3.2123 6 3.2123 6 3.21237 3.9864 7 3.9864 7 3.9864 7 4.2334 7 4.2334 7 4.23348 4.9715 8 4.9715 8 4.9715 8 5.376 8 5.376 8 5.3769 5.7351 9 5.7351 9 5.7351 9 6.6302 9 6.6302 9 6.630210 6.7297 10 6.7297 10 6.7297 10 7.892 10 7.892 10 7.892
Cantiliver beam Analytical Natural Frequencies, Euler-Bernoullin knl wn
E 30000 Pa 1 1.875 2.291h 0.1 m 2 4.696 14.370b 0.1 m 3 7.855 40.205L 0.4 m 4 10.996 78.781
h/L 0.250 5 14.137 130.230Ro 2300 kg/m3 6 17.279 194.541I 8.333333E-06 m4 7 20.420 271.714
P2 0.65161 8 23.562 361.750L/h 4 9 26.704 464.648
10 29.845 580.408
10-linear 10-quadratic 20-linear73.961 73.6620 0.405% 74.23 -0.363%
463.937 447.2700 3.593% 450.64 2.866%1298.062 1203.8000 7.262% 1206.4 7.061%2543.542 2243.1000 11.812% 2234.1 12.166%4204.631 4887.2000 -16.234% 4876.1 -15.970%6280.992 6297.2000 -0.258% 6414.2 -2.121%8772.625 7591.4000 13.465% 8074.1 7.963%
11679.530 11973 -2.513%15001.708 14957.1 0.297%18739.157
3D SOLID Linear -Abaqus
Analytical Natural Frequencies, Euler-Bernoulli 10 20 30f(Hz)
0.3646 0.34042.2870 1.74966.3988 4.0295
12.5384 6.462820.726730.962243.244757.574373.951092.3747
3D Solid Quadratic-Abaqus
10 20 30
