A Chebyshev Spectral Method with Null Space Approach for Boundary-Value Problems of Euler-Bernoulli Beam

Table 6

The first four nondimensional natural frequencies for clamped-free and simply supported beams with a step ratio ς = 0.5.

BCs	Mode index	Beam type	Methods	Length ratio of the first segment (R1/l)
				0.25	0.375	0.625	0.75
Clamped-free	1	Beam-A	Present	2.0849	2.1526	2.1526	2.0849
			Ref. [9]	2.0849	2.1526	2.1526	2.0849
		Beam-B	Present	1.6687	1.8696	2.1193	2.0813
			Ref. [9]	1.6687	1.8696	2.1193	2.0813
		Beam-C	Present	1.7151	1.9777	2.3651	2.2454
			Ref. [9]	1.7151	1.9777	2.3651	2.2454
	2	Beam-A	Present	4.9153	4.7950	4.7950	4.9153
			Ref. [9]	4.9153	4.7950	4.7950	4.9153
		Beam-B	Present	3.9528	3.9387	4.0976	4.6760
			Ref. [9]	3.9528	3.9387	4.0976	4.6760
		Beam-C	Present	4.1395	4.1638	3.9885	4.8251
			Ref. [9]	4.1395	4.1638	3.9885	4.8521
	3	Beam-A	Present	7.9045	7.8343	7.8343	7.9045
			Ref. [9]	7.9045	7.8343	7.8343	7.9045
		Beam-B	Present	6.1635	6.1399	6.8479	6.9380
			Ref. [9]	6.1635	6.1399	6.8479	6.9380
		Beam-C	Present	6.3841	6.0002	7.0045	6.7812
			Ref. [9]	6.8341	6.0002	7.0045	6.7812
	4	Beam-A	Present	10.9711	11.0414	11.0414	10.9711
			Ref. [9]	10.9711	11.0414	11.0414	10.9711
		Beam-B	Present	8.3054	8.8732	9.4841	9.9945
			Ref. [9]	8.3054	8.8733	9.4841	9.9945
		Beam-C	Present	8.2156	8.8775	9.3342	10.1310
			Ref. [9]	8.2156	8.8775	9.3342	10.1310
Simply supported	1	Beam-A	Present	3.1160	3.0890	3.0890	3.1160
			Ref. [9]	3.1160	3.0890	3.0890	3.1160
		Beam-B	Present	2.2270	2.2637	2.5443	2.8238
			Ref. [9]	2.2270	2.2637	2.5443	2.8238
		Beam-C	Present	2.1602	2.1257	2.3111	2.6160
			Ref. [9]	2.1602	2.1257	2.3111	2.6160
	2	Beam-A	Present	6.2218	6.2831	6.2831	6.2218
			Ref. [9]	6.2218	6.2831	6.2831	6.2218
		Beam-B	Present	4.5706	4.9160	5.5403	5.4805
			Ref. [9]	4.5706	4.9160	5.5403	5.4805
		Beam-C	Present	4.3995	4.7788	5.7147	5.4039
			Ref. [9]	4.3995	4.7788	5.7147	5.4039
	3	Beam-A	Present	9.4248	9.4667	9.4667	9.4248
			Ref. [9]	9.4248	9.4667	9.4667	9.4248
		Beam-B	Present	7.0984	7.6854	8.1049	8.6897
			Ref. [9]	7.0984	7.6854	8.1049	8.6897
		Beam-C	Present	6.9661	7.7817	7.9441	8.8739
			Ref. [9]	6.9661	7.7817	7.9441	8.8739
	4	Beam-A	Present	12.6251	12.5075	12.5075	12.6251
			Ref. [9]	12.6253	12.5076	12.5076	12.6253
		Beam-B	Present	9.6906	9.9389	10.8726	11.4763
			Ref. [9]	9.6908	9.9390	10.8727	11.4765
		Beam-C	Present	9.6683	10.0408	10.9903	11.4005
			Ref. [9]	9.6684	10.0409	10.9905	11.4007