Analytical Solution for the Free Vibration Analysis of Delaminated Timoshenko Beams

Table 2

First three dimensionless natural frequencies of the beam with a central and midplane delamination.

	Method	𝐿2
		0.001	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
Ω1	Present	22.39	21.48	22.20	21.34	21.69	21.36	19.91	17.68	15.25	13.06
	Analytical [6]	22.37	22.37	22.36	22.24	21.83	20.89	19.30	17.23	15.05	13.00
	Analytical [1]	22.39	22.37	22.35	22.23	21.83	20.88	19.29	17.23	15.05	13.00
	FEM [16]	22.36	22.36	22.35	22.23	21.82	20.88	19.28	17.22	15.05	12.99
Ω2	Present	61.66	60.79	55.98	48.99	43.93	41.52	40.98	40.81	39.06	35.39
	Analytical [6]	61.67	60.81	56.00	49.00	43.89	41.52	41.04	40.82	39.07	35.39
	Analytical [1]	61.67	60.76	55.97	49.00	43.87	41.45	40.93	40.72	39.01	35.38
	FEM [16]	61.61	60.74	55.95	48.97	43.86	41.50	41.01	40.80	39.04	35.38
Ω3	Present	120.75	115.13	118.02	107.22	93.58	82.36	78.28	78.61	76.57	69.50
	Analytical [6]	120.90	120.83	118.87	109.16	93.59	82.29	77.69	77.18	75.43	69.19
	Analytical [1]	120.91	120.81	118.76	109.04	93.57	82.29	77.64	77.05	75.33	69.17
	FEM [16]	120.68	120.62	118.69	109.03	93.51	82.23	77.64	77.12	75.39	69.16