Free Transverse Vibration Analysis of Axially Functionally Graded Tapered Euler-Bernoulli Beams through Spline Finite Point Method

Table 10

Nondimensional higher-order transverse natural frequencies for axially FG beams with respect to the gradient index.

		P-P		C-C		C-P		C-F
		Case  1	Case  2	Case  1	Case  2	Case  1	Case  2	Case  1	Case  2
−10		45.6887	40.0700	68.1328	66.2401	54.9276	53.9917	23.7075	24.4362
		102.6601	90.5367	135.4215	127.6871	116.4430	110.4619	68.1642	66.2609
		182.4345	161.3146	225.7493	209.1345	201.0000	186.8845	135.4187	127.6897
		285.1358	252.4647	339.2851	310.8793	308.7099	283.5206	225.7516	209.1407
−3		44.8550	41.9691	67.7638	67.1081	54.4731	54.9320	23.0531	25.4924
		100.8244	94.5153	134.4815	130.2588	115.5467	112.8131	67.4843	67.7907
		179.1822	168.0682	223.7508	214.4078	199.1799	191.6794	134.1969	130.8653
		280.0397	262.7506	335.7204	319.7813	305.4568	291.7087	223.4883	214.9894
0		43.6649	43.6649	67.5918	67.5918	53.9610	55.9731	22.3541	26.4650
		98.1296	98.1296	132.8706	132.8706	113.7904	115.8588	66.1736	70.2707
		174.3787	174.3787	220.0032	220.0032	195.4598	197.5581	131.4575	135.6415
		272.5370	272.5370	329.1761	329.1761	299.1037	301.2218	218.6135	222.8467
3		41.9691	44.8550	67.1081	67.7638	52.8041	56.9021	21.4951	27.1414
		94.5153	100.8244	130.2588	134.4815	110.6243	118.4554	63.6755	72.7816
		168.0682	179.1822	214.4078	223.7508	189.4610	202.4128	126.6073	140.3662
		262.7506	280.0397	319.7813	335.7204	289.4704	308.9241	210.6519	230.2512
10		40.0700	45.6887	66.2401	68.1328	50.7956	57.7893	20.7361	27.0569
		90.5367	102.6601	127.6871	135.4215	106.3245	120.4638	60.1871	74.3559
		161.3146	182.4345	209.1345	225.7493	182.1851	205.9533	120.0294	144.0360
		252.4647	285.1358	310.8793	339.2851	278.4567	314.4208	200.4314	236.3716