Table 2 The first five frequencies and nonlinear correction term for various γ and κ values for clamped-clamped 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 | |
|---|---|---|---|---|---|---|---|---|
Clamped-Clamped | 1 | 0 | 22.3956 | 61.6809 | 120.908 | 199.862 | 298.557 | 1.26716 |
0.1 | 21.1327 | 50.993 | 85.7223 | 121.352 | 156.744 | 1.45493 | ||
0.2 | 18.3167 | 36.4377 | 54.5331 | 71.6196 | 88.4926 | 1.73402 | ||
0.3 | 15.3861 | 27.0202 | 38.8468 | 49.687 | 60.7786 | 2.30212 | ||
0.4 | 12.9434 | 21.1635 | 29.9791 | 37.841 | 46.1611 | 3.33512 | ||
0.5 | 11.0368 | 17.3019 | 24.3529 | 30.501 | 37.1779 | 4.68740 | ||
10 | 0 | 22.5957 | 61.7538 | 120.945 | 199.884 | 298.572 | 1.25595 | |
0.1 | 21.3446 | 51.0811 | 85.7747 | 121.389 | 156.772 | 1.44049 | ||
0.2 | 18.5608 | 36.5609 | 54.6156 | 71.6824 | 88.5434 | 1.71122 | ||
0.3 | 15.6759 | 27.1862 | 38.9625 | 49.7775 | 60.8526 | 2.25956 | ||
0.4 | 13.2865 | 21.375 | 30.1288 | 37.9597 | 46.2584 | 3.24900 | ||
0.5 | 11.4372 | 17.5601 | 24.537 | 30.6482 | 37.2987 | 4.52332 | ||
50 | 0 | 23.4641 | 62.0769 | 121.11 | 199.984 | 298.639 | 1.20946 | |
0.1 | 22.2619 | 51.4712 | 86.0076 | 121.554 | 156.9 | 1.38114 | ||
0.2 | 19.6087 | 37.1039 | 54.9806 | 71.9608 | 88.769 | 1.61976 | ||
0.3 | 16.9036 | 27.9122 | 39.4725 | 50.1777 | 61.1804 | 2.09545 | ||
0.4 | 14.715 | 22.2911 | 30.7855 | 38.483 | 46.6888 | 2.93359 | ||
0.5 | 13.0694 | 18.6643 | 25.339 | 31.2939 | 37.8312 | 3.95841 | ||
100 | 0 | 24.5064 | 62.4783 | 121.316 | 200.109 | 298.723 | 1.15802 | |
0.1 | 23.3579 | 51.9546 | 86.2978 | 121.759 | 157.059 | 1.31633 | ||
0.2 | 20.8447 | 37.7717 | 55.4334 | 72.3074 | 89.0502 | 1.52372 | ||
0.3 | 18.323 | 28.7939 | 40.1008 | 50.6734 | 61.5876 | 1.93312 | ||
0.4 | 16.3258 | 23.3857 | 31.5871 | 39.1272 | 47.2212 | 2.64414 | ||
0.5 | 14.8597 | 19.9588 | 26.3071 | 32.0829 | 38.4863 | 3.48149 | ||
200 | 0 | 26.4682 | 63.2735 | 121.728 | 200.359 | 298.89 | 1.07219 | |
0.1 | 25.4085 | 52.9083 | 86.8752 | 122.169 | 157.377 | 1.21010 | ||
0.2 | 23.1193 | 39.073 | 56.3282 | 72.9956 | 89.6099 | 1.37381 | ||
0.3 | 20.8742 | 30.481 | 41.3289 | 51.6507 | 62.3942 | 1.69686 | ||
0.4 | 19.145 | 25.4341 | 33.1323 | 40.3849 | 48.2685 | 2.25478 | ||
0.5 | 17.9112 | 22.3239 | 28.1437 | 33.6052 | 39.7643 | 2.88835 | ||
500 | 0 | 31.6317 | 65.6013 | 122.954 | 201.106 | 299.392 | 0.89717 | |
0.1 | 30.7505 | 55.6712 | 88.585 | 123.391 | 158.327 | 0.99988 | ||
0.2 | 28.8878 | 42.7399 | 58.931 | 75.0224 | 91.2685 | 1.09948 | ||
0.3 | 27.1244 | 35.0584 | 44.8116 | 54.4775 | 64.7537 | 1.30586 | ||
0.4 | 25.8173 | 30.7716 | 37.3865 | 43.9424 | 51.282 | 1.67204 | ||
0.5 | 24.9161 | 28.2552 | 33.0464 | 37.8062 | 43.3728 | 2.07632 |