PP |
-10 |
Present |
9.9225 |
39.8335 |
89.3251 |
157.4965 |
Huang and Li (2010Huang, Y., Li, X.F. (2010). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J Sound Vib 329:2291-303.) |
9.9358 |
- |
- |
- |
Liu et al. (2016Liu, P., Lin, K., Liu, H., Qin, R. (2016). Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock Vib 2016: Article ID 5891030, 23 pages.) |
9.9366 |
40.07 |
90.5367 |
161.3146 |
-3 |
Present |
10.3513 |
41.7155 |
93.2389 |
164.0748 |
Huang and Li (2010Huang, Y., Li, X.F. (2010). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J Sound Vib 329:2291-303.) |
10.3669 |
- |
- |
- |
Liu et al. (2016Liu, P., Lin, K., Liu, H., Qin, R. (2016). Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock Vib 2016: Article ID 5891030, 23 pages.) |
10.3669 |
41.9691 |
94.5153 |
168.0682 |
0 |
Present |
10.8486 |
43.3957 |
96.7930 |
170.2173 |
Huang and Li (2010Huang, Y., Li, X.F. (2010). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J Sound Vib 329:2291-303.) |
10.8663 |
- |
- |
- |
Liu et al. (2016Liu, P., Lin, K., Liu, H., Qin, R. (2016). Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock Vib 2016: Article ID 5891030, 23 pages.) |
10.8664 |
43.6649 |
98.1296 |
174.3787 |
3 |
Present |
11.2250 |
44.5707 |
99.4306 |
174.8680 |
Huang and Li (2010Huang, Y., Li, X.F. (2010). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J Sound Vib 329:2291-303.) |
11.2443 |
- |
- |
- |
Liu et al. (2016Liu, P., Lin, K., Liu, H., Qin, R. (2016). Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock Vib 2016: Article ID 5891030, 23 pages.) |
11.2443 |
44.855 |
100.8244 |
179.1822 |
10 |
Present |
11.4367 |
45.3939 |
101.2149 |
177.9781 |
Huang and Li (2010Huang, Y., Li, X.F. (2010). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J Sound Vib 329:2291-303.) |
11.4532 |
- |
- |
- |
Liu et al. (2016Liu, P., Lin, K., Liu, H., Qin, R. (2016). Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock Vib 2016: Article ID 5891030, 23 pages.) |
11.4560 |
45.6887 |
102.6601 |
182.4345 |
CC |
-10 |
Present |
24.6463 |
65.2907 |
124.5252 |
201.2268 |
Huang and Li (2010Huang, Y., Li, X.F. (2010). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J Sound Vib 329:2291-303.) |
24.7949 |
- |
- |
- |
Liu et al. (2016Liu, P., Lin, K., Liu, H., Qin, R. (2016). Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock Vib 2016: Article ID 5891030, 23 pages.) |
24.8068 |
66.2401 |
127.6871 |
209.1345 |
-3 |
Present |
24.7797 |
66.1549 |
127.0355 |
206.2868 |
Huang and Li (2010Huang, Y., Li, X.F. (2010). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J Sound Vib 329:2291-303.) |
24.9375 |
- |
- |
- |
Liu et al. (2016Liu, P., Lin, K., Liu, H., Qin, R. (2016). Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock Vib 2016: Article ID 5891030, 23 pages.) |
24.9366 |
67.1081 |
130.2588 |
214.4078 |
0 |
Present |
24.2226 |
66.6214 |
129.5533 |
211.6095 |
Huang and Li (2010Huang, Y., Li, X.F. (2010). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J Sound Vib 329:2291-303.) |
24.3752 |
- |
- |
- |
Liu et al. (2016Liu, P., Lin, K., Liu, H., Qin, R. (2016). Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock Vib 2016: Article ID 5891030, 23 pages.) |
24.3756 |
67.5918 |
132.8706 |
220.0032 |
3 |
Present |
23.7942 |
66.7804 |
131.0912 |
215.1447 |
Huang and Li (2010Huang, Y., Li, X.F. (2010). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J Sound Vib 329:2291-303.) |
23.9456 |
- |
- |
- |
Liu et al. (2016Liu, P., Lin, K., Liu, H., Qin, R. (2016). Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock Vib 2016: Article ID 5891030, 23 pages.) |
23.9446 |
67.7638 |
134.4815 |
223.7508 |
10 |
Present |
23.9070 |
67.1197 |
131.9397 |
216.9219 |
Huang and Li (2010Huang, Y., Li, X.F. (2010). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J Sound Vib 329:2291-303.) |
24.0576 |
- |
- |
- |
Liu et al. (2016Liu, P., Lin, K., Liu, H., Qin, R. (2016). Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method, Shock Vib 2016: Article ID 5891030, 23 pages.) |
24.0660 |
68.1328 |
135.4215 |
225.7493 |