An Investigation on the Nonlinear Free Vibration Analysis of Beams with Simply Supported Boundary Conditions Using Four Engineering Theories

Table 1

Comparison of the nondimensional nonlinear frequency (𝜔_NL).

𝑎	𝜂	Reference [6]	EBBM^2	RBM^3	SBM^4	TBM^5
						(𝜔NL1)	(𝜔NL2)
0.4	10	8.5458(4)^1	10.017	9.5561	8.8200	8.5452	67.412
	25	9.7198(3)		9.9384	9.7906	9.7200	369.04
	50	9.9391(3)		9.9968	9.9585	9.9393	1442.8
	100	9.9968(3)		10.012	10.002	9.9970	5737.3
0.8	10	9.0018(6)	10.445	9.9648	9.3037	9.0103	67.438
	25	10.1494(5)		10.364	10.228	10.155	369.05
	50	10.3650(4)		10.424	10.389	10.369	1442.8
	100	10.4217(4)		10.440	10.431	10.426	5737.3
1.0	10	9.3200(7)	10.755	10.261	9.6506	9.3436	67.458
	25	10.4563(5)		10.671	10.545	10.469	369.05
	50	10.6702(5)		10.734	10.701	10.680	1442.8
	100	10.7265(5)		10.750	10.742	10.736	5737.3
2.0	10	11.4868(16)	13.056	12.456	12.163	11.747	67.623
	25	12.6458(8)		12.954	12.884	12.790	369.08
	50	12.8689(7)		13.031	13.012	12.987	1442.9
	100	12.9279(7)		13.050	13.045	13.039	5737.3
3.0	10	14.1558(47)	16.180	15.436	15.468	14.876	67.907
	25	15.4532(14)		16.054	16.041	15.922	369.12
	50	15.7178(11)		16.148	16.144	16.113	1442.9
	100	15.7889(11)		16.172	16.171	16.163	5737.3

¹Numbers in parentheses indicate the number of iterations required to achieve an accuracy of 10⁻⁶ in the evaluation of the nonlinear- to-linear frequency ratio.
²EBBM: Bernoulli-Euler beam model.
³RBM: Rayleigh beam model.
⁴SBM: shear beam model.
⁵TBM: Timoshenko beam model.