Table 1 The first five frequencies and nonlinear correction term for various γ and κ values for simple-simple support condition
From: Nonlocal beam theory for nonlinear vibrations of a nanobeam resting on elastic foundation
κ | γ | \(\boldsymbol{\omega_{1}}\) | \(\boldsymbol{\omega_{2}}\) | \(\boldsymbol{\omega_{3}}\) | \(\boldsymbol{\omega_{4}}\) | \(\boldsymbol{\omega_{5}}\) | λ | |
|---|---|---|---|---|---|---|---|---|
Simple-Simple | 1 | 0 | 9.9201 | 39.4911 | 88.8321 | 157.917 | 246.742 | 1.84113 |
0.1 | 9.46883 | 33.4426 | 64.6491 | 98.3343 | 132.51 | 1.93244 | ||
0.2 | 8.41654 | 24.6026 | 41.6405 | 58.3889 | 74.8465 | 2.23004 | ||
0.3 | 7.25166 | 18.5286 | 29.6349 | 40.5001 | 51.2291 | 2.77290 | ||
0.4 | 6.2264 | 14.6293 | 22.7963 | 30.8283 | 38.7947 | 3.52508 | ||
0.5 | 5.39378 | 12.0161 | 18.4661 | 24.8405 | 31.1804 | 4.40404 | ||
10 | 0 | 10.3638 | 39.6049 | 88.8827 | 157.945 | 246.76 | 1.76231 | |
0.1 | 9.93271 | 33.5769 | 64.7187 | 98.38 | 132.544 | 1.84219 | ||
0.2 | 8.93522 | 24.7849 | 41.7484 | 58.4659 | 74.9066 | 2.10059 | ||
0.3 | 7.84771 | 18.7699 | 29.7864 | 40.611 | 51.3169 | 2.56230 | ||
0.4 | 6.91145 | 14.9337 | 22.9928 | 30.9739 | 38.9105 | 3.17568 | ||
0.5 | 6.17194 | 12.3849 | 18.7082 | 25.021 | 31.3244 | 3.84878 | ||
50 | 0 | 12.1412 | 40.1067 | 89.1074 | 158.072 | 246.841 | 1.50432 | |
0.1 | 11.7753 | 34.1674 | 65.027 | 98.5831 | 132.695 | 1.55392 | ||
0.2 | 10.9471 | 25.5791 | 42.2248 | 58.807 | 75.1732 | 1.71454 | ||
0.3 | 10.079 | 19.8068 | 30.4504 | 41.1005 | 51.7051 | 1.99506 | ||
0.4 | 9.36846 | 16.2178 | 23.8468 | 31.6131 | 39.4212 | 2.34281 | ||
0.5 | 8.83701 | 13.9063 | 19.7483 | 25.8079 | 31.9565 | 2.68806 | ||
100 | 0 | 14.0502 | 40.7252 | 89.3876 | 158.23 | 246.943 | 1.29992 | |
0.1 | 13.7353 | 34.8914 | 65.4103 | 98.8364 | 132.883 | 1.33218 | ||
0.2 | 13.0322 | 26.5385 | 42.8127 | 59.2306 | 75.505 | 1.44022 | ||
0.3 | 12.312 | 21.0311 | 31.2607 | 41.7044 | 52.1864 | 1.63322 | ||
0.4 | 11.7375 | 17.6923 | 24.8731 | 32.3942 | 40.0503 | 1.86995 | ||
0.5 | 11.3178 | 15.6008 | 20.9761 | 26.7591 | 32.7294 | 2.09886 | ||
200 | 0 | 17.2456 | 41.935 | 89.9452 | 158.546 | 247.145 | 1.05906 | |
0.1 | 16.99 | 36.2961 | 66.1703 | 99.341 | 133.259 | 1.07698 | ||
0.2 | 16.4267 | 28.36 | 43.9651 | 60.0688 | 76.1643 | 1.14261 | ||
0.3 | 15.8615 | 23.2875 | 32.8212 | 42.8865 | 53.1359 | 1.26773 | ||
0.4 | 15.4197 | 20.3228 | 26.808 | 33.9026 | 41.2799 | 1.42342 | ||
0.5 | 15.1027 | 18.5307 | 23.2378 | 28.5666 | 34.223 | 1.57287 | ||
500 | 0 | 24.442 | 45.3712 | 91.5977 | 159.489 | 247.751 | 0.74725 | |
0.1 | 24.2623 | 40.217 | 68.3997 | 100.84 | 134.38 | 0.75417 | ||
0.2 | 23.8713 | 33.2309 | 47.2539 | 62.5161 | 78.1089 | 0.78627 | ||
0.3 | 23.4859 | 29.0225 | 37.111 | 46.2521 | 55.8876 | 0.85618 | ||
0.4 | 23.1898 | 26.7024 | 31.9166 | 38.0708 | 51.7907 | 0.94648 | ||
0.5 | 22.9803 | 25.365 | 28.9827 | 33.4073 | 38.3564 | 1.03368 |