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

Table 5

The first three dimensionless natural frequencies of a cone beam with tip-mass, the rotatory inertia of mass, and its eccentricity at the left end (α_b=α_h=α=1.1, β₁=0, β₂=0, and μ_R = δ_R = γ_R = δ_L =0).

δL	β4	γL	β3
				Present	(Ref. [7])	Present	(Ref. [7])	Present	(Ref. [7])
0.4	∞	0.6	0.1	0.467442	0.467440	1.854357	1.854354	4.358113	4.358113
			1.0	0.755242	0.755247	1.948854	1.948854	4.454283	4.454283
			10.0	0.934564	0.934568	2.182581	2.182581	4.834589	4.834589
		1.0	0.1	0.441666	0.441668	1.583464	1.583463	4.256809	4.256810
			1.0	0.703336	0.703347	1.685049	1.685046	4.358454	4.358455
			10.0	0.850072	0.850076	1.921416	1.921416	4.751734	4.751734
	1.0	0.6	0.1	0.461904	0.461893	1.165267	1.165266	2.480518	2.480518
			1.0	0.700856	0.700855	1.197688	1.197686	2.634190	2.634190
			10.0	0.801466	0.801468	1.238414	1.238412	2.891271	2.891271
0.6	∞	0.6	0.1	0.443601	0.443598	1.873050	1.873048	4.436802	4.436802
			1.0	0.712292	0.712291	1.983124	1.983125	4.530416	4.530416
			10.0	0.873798	0.873804	2.248240	2.248241	4.905603	4.905603
		1.0	0.1	0.423210	0.423220	1.621962	1.621961	4.295470	4.295470
			1.0	0.671928	0.671929	1.732488	1.732486	4.395387	4.395387
			10.0	0.809967	0.809974	1.984642	1.984642	4.784972	4.784972
	1.0	0.6	0.1	0.439333	0.439333	1.162506	1.162500	2.520126	2.520126
			1.0	0.671066	0.671065	1.184763	1.184762	2.684893	2.684893
			10.0	0.773950	0.773954	1.212627	1.212624	2.960297	2.960296